Extensions of flat functors and theories of presheaf type
aa r X i v : . [ m a t h . C T ] J un Extensions of flat functorsand theories of presheaf type
Olivia Caramello ∗ Institut des Hautes Études Scientifiques35 route de Chartres 91440, Bures-sur-Yvette, [email protected]
June 20, 2014
Abstract
We develop a general theory of extensions of flat functors alonggeometric morphisms of toposes, and apply it to the study of the classof theories whose classifying topos is equivalent to a presheaf topos.As a result, we obtain a characterization theorem providing necessaryand sufficient semantic conditions for a theory to be of presheaf type.This theorem subsumes all the previous partial results obtained on thesubject and has several corollaries which can be used in practice fortesting whether a given theory is of presheaf type as well as for gener-ating new examples of theories belonging to this class. Along the way,we establish a number of other results of independent interest, includ-ing developments about colimits in the context of indexed categories,expansions of geometric theories and methods for constructing theoriesclassified by a given presheaf topos.
Contents E -filtered indexed categories . . . . . . . . . . . . . . . . . . . 102.3 Indexation of internal diagrams . . . . . . . . . . . . . . . . . 102.4 Colimits and tensor products . . . . . . . . . . . . . . . . . . 112.5 E -final subcategories . . . . . . . . . . . . . . . . . . . . . . . 17 ∗ Supported by a CARMIN IHÉS-IHP post-doctoral position (as from 1/12/2013)and by a visiting position of the Max Planck Institute for Mathematics (in the period1/10/2013 - 30/11/2013). E -indexed colimits . . . . . . . . . . . . 242.7 Explicit calculation of set-indexed colimits . . . . . . . . . . . 36 E -finite presentability . . . . . . . . . . . . . . . . . 69 ( i ) . . . . . . . . . . . . . . . . . . . . . . . 775.2.2 Condition ( ii ) . . . . . . . . . . . . . . . . . . . . . . . 815.2.3 Condition ( iii ) . . . . . . . . . . . . . . . . . . . . . . 845.3 Abstract reformulation . . . . . . . . . . . . . . . . . . . . . . 93 ( iii ) of Theorem 5.1 . . . . . . . 1026.4 Quotient theories . . . . . . . . . . . . . . . . . . . . . . . . . 1066.4.1 Presheaf-type quotients and rigid topologies . . . . . . 1086.4.2 Finding theories classified by a given presheaf topos . 1116.5 Injectivizations of theories . . . . . . . . . . . . . . . . . . . . 1156.5.1 Condition (iii) of Theorem 5.1 for injectivizations . . . 1186.5.2 A criterion for injectivizations . . . . . . . . . . . . . . 122 l -groups with strong unit . . . . . . . . 155 Following [2], we say that a geometric theory is of of presheaf type if it isclassified by a presheaf topos.A geometric theory T is of presheaf type if and only if it is classifiedby the topos [ f.p. T -mod ( Set ) , Set ] , where f.p. T -mod ( Set ) is (a skeleton of)the full subcategory of T -mod ( Set ) on the finitely presentable T -models (cf.[5]).The subject of theories of presheaf type has a long history, starting withthe book [19] by Hakim, which first introduced the point of view of classifyingtoposes in the context of the theory of commutative rings with unit andits quotients. The subsequent pionereeing work [25] by Lawvere led to thediscovery that any finitary algebraic theory is of presheaf type, classified bythe topos of presheaves on the opposite of its category of finitely presentablemodels (cf. [23]). This result was later generalized to cartesian (or essentiallyalgebraic) theories as well as to universal Horn theories (cf. [3]). At the sametime, new examples of non-cartesian theories of presheaf type were discovered(cf. for instance [2] for a long, but by no means exhaustive, list of examples),and partial results in connection to the problem of characterizing the classof theories of presheaf type emerged; for instance, [22], [2] and [32] containdifferent sets of sufficient conditions for a theory to be of presheaf type.Theories of presheaf type occupy a central role in Topos Theory for anumber of reasons:(i) Every small category C can be seen, up to Cauchy-completion, as thecategory of finitely presentable models of a theory of presheaf type(namely, the theory of flat functors on C op );(ii) As every Grothendieck topos is a subtopos of some presheaf topos, soevery geometric theory is a quotient of some theory of presheaf type(cf. the duality theorem of [9] between subtoposes of the classifyingtopos of a geometric theory and quotients of the theory);(iii) Every finitary algebraic theory (and more generally, any cartesian the-ory) is of presheaf type;(iv) The class of theories of presheaf type contains, besides all cartesiantheories, many other interesting mathematical theories pertaining todifferent fields of mathematics (for instance, the coherent theory of3inear orders or the geometric theory of algebraic extensions of a givenfield);(v) The ‘bridge technique’ of [10] can be fruitfully applied in the contextof theories of presheaf type due to the fact that the classifying topos ofany such theory admits (at least) two quite different representations,one of semantic nature (namely, set-valued functors on the category offinitely presentable models of the theory) and one of syntactic nature(namely, sheaves on the syntactic site of the theory).It is therefore important to dispose of effective criteria for testing whethera theory is of presheaf type, as well as of methods for generating new theoriesof presheaf type.In this paper, we carry out a systematic investigation of this class of the-ories, obtaining in particular a characterization theorem providing necessaryand sufficient conditions for a theory to be of presheaf type, expressed interms of the models of the theory in arbitrary Grothendieck toposes. Thistheorem, whose general statement is quite abstract, admits several ramifi-cations and simpler corollaries which can be effectively applied in practiceto test whether a given theory is of presheaf type as well as for generatingnew examples of theories of presheaf type, also through appropriate ‘modi-fications’ of given geometric theories. All the partial results and recognitioncriteria previously obtained on the subject are naturally subsumed by thisgeneral result; moreover, the constructive nature of the characterization the-orem allows to replace the requirements that the theory should have enoughset-based models in the sufficient criteria of [2] and [22] with explicit seman-tic conditions which can be directly verified without having to invoke anyform of the axiom of choice.In order to establish our characterization theorem, we embark, in the firsttwo sections of the paper, in a general analysis of indexed colimits in toposesand extensions of flat functors along geometric morphisms. In fact, the nec-essary and sufficient conditions for a theory T to be of presheaf type providedby the characterization theorem arise precisely from the requirement that forany Grothendieck topos E , the operation of extension of flat functors withvalues in E from the opposite of the category of finitely presentable modelsof T to the geometric syntactic category of T should define an equivalenceof categories onto the category of T -models in E , naturally in E . We theninvestigate the preservation, by ‘faithful interpretations’ of theories, of eachof the conditions in the characterization theorem, obtaining results of theform ‘under appropriate conditions, a geometric theory in which a theoryof presheaf type faithfully interprets is again of presheaf type’. Finally, wediscuss known and new examples of theories of presheaf type in light of thetheory developed in the paper.More specifically, the contents of the paper can be summarized as follows.4n section 2, we investigate E -indexed colimits of internal diagrams inGrothendieck toposes E , analyzing in particular their behaviour with re-spect to final E -indexed functors (cf. section 2.5) and establishing explicitcharacterizations for a E -indexed cocone to be colimiting (cf. section 2.6).In section 2.4, we exploit the abstract interpretation of colimits as kinds oftensor products to derive commutation results which play an important rolein the subsequent parts of the paper as they allow us to interpret certain set-indexed colimits arising in the context of our main characterization theoremas special kinds of filtered indexed colimits.In section 3, we investigate the properties of the operation on flat func-tors induced by a geometric morphism of toposes via Diaconescu’s equiv-alence. We focus in particular on geometric morphisms between presheaftoposes induced by embeddings between small categories, and on geomet-ric morphisms to the classifying topos of a geometric theory induced by asmall category of set-based models of the theory. We also establish, in sec-tion 3.4, a general ‘hom-tensor’ adjunction between categories of E -valuedfunctors (for E a Grothedieck topos) [ C , E ] and [ D , E ] induced by a functor P : C → [ D op , Set ] , which generalizes the well-known adjunction induced byKan extensions along a given functor.In section 4, in order to set up the field for the statement and proof ofthe characterization theorem, we identify some notable properties of theoriesof presheaf type, notably including the fact that every finitely presentablemodel of such a theory is finitely presented - in a strong sense which we makeprecise in section 4.3 - and admits an entirely syntactic description in terms ofthe signature of the theory and the notion of provability of geometric sequentsover it in the theory (cf. section 4.1). We also show, in section 4.2, that forany geometric theory T and any Grothendieck topos E , there exists for eachpair of T -models M and N in E , an ‘object of T -model homomorphisms’ in E from M to N which classifies the T -model homomorphisms in slices of E between the localizations of M and N in it.In section 5, we establish our main characterization theorem providingnecessary and sufficient conditions for a geometric theory to be classifiedby a presheaf topos. We first state the result abstractly and then proceedto obtain explicit reformulations of each of the conditions. We also derivesome corollaries which allow to verify the satisfaction of the conditions ofthe theorem in specific situations which naturally arise in practice. Lastly,we show that, once recast in the language of indexed colimits of internaldiagrams in toposes, the conditions of the characterization theorem for agiven geometric theory T amount precisely to requirement that every modelof T in any Grothendieck topos E should be a canonical E -indexed colimitof a certain E -filtered diagram of ‘constant’ finitely presentable models of T which are E -finitely presentable.In section 6, we introduce the notion of faithful interpretation of geomet-ric theories and investigate to what extent the satisfaction of the conditions of5he characterization theorem is preserved by this kind of interpretations. Asapplications of the general results that we obtain on this topic, we considerin particular the case of quotients of a given geometric theory T and thatof injectivizations (i.e., theories obtained by adding, for each sort over thesignature of the theory, a binary predicate which is provably complementedto the equality relation relative to that sort), providing various sufficientconditions for these theories to be of presheaf type if T is. To this end, wecarry out in section 6.2 a general analysis of the relationship between finitelypresentable and finitely generated models of a given geometric theory. Insection 6.4.2, we treat the problem of finding a geometric theory classifiedby a given presheaf topos [ K , Set ] , and prove a general theorem ensuringthat if the category K can be realized as a full subcategory of the categoryof finitely presentable models of a theory of presheaf type T , there exists aquotient of T classified by the topos [ K , Set ] , which can be described in mostexplicit ways in terms of T and K provided that some natural conditions aresatisfied. We also discuss, in section 6.4.1, the relationship between rigidtopologies on the opposite of the category of finitely presentable models ofa theory of presheaf type T and the presheaf-type quotients of T .In section 7, we investigate expansions of geometric theories from thepoint of view of the geometric morphisms that they induce between therespective classifying toposes. In particular, we introduce the notion of alocalic (resp. hyperconnected) expansion, and show that it naturally corre-sponds to the notion of localic (resp. hyperconnected) geometric morphismat the level of classifying toposes; as a result, we obtain a logical characteri-zation of the hyperconnected-localic factorization of a geometric morphism.Next, we address the problem of expanding a given geometric theory T toa theory classified by a presheaf topos of the form [ K , Set ] , where K is asmall category of set-based models of T , and describe a general method fordefining such expansions.In section 8, we discuss classical, as well as new, examples of theories ofpresheaf type from the perspective of the theory developed in the paper. Werevisit in particular well-known examples of theories of presheaf type whosefinitely presentable models are all finite, notably including the geometrictheory of finite sets, and give fully constructive proofs of the fact that Mo-erdijk’s theory of abstract circles and Johnstone’s theory of Diers fields areof presheaf type. Next, we introduce new examples of theories of presheaftype, including the theory of algebraic extensions of a given field, the theoryof locally finite groups, the theory of vector spaces with linear independencepredicates and the theory of abelian ℓ -groups with strong unit. We also showthat the injectivization of the algebraic theory of groups is not of presheaftype and explicitly describe a presheaf completion for it.6 Indexed colimits in toposes
Before proceeding further, we need to recall some standard notions and factsfrom the theory of indexed categories; we refer the reader to [21] (especiallysections B1.2, B2.3 and B3) and to [29] for the background.Given an internal category C in a topos E , we denote by C its object ofarrows, by C its object of objects and by d C , d C : C → C the domain andcodomain arrows.Given a small category C and a topos E defined over Set , we can alwaysinternalize C into E by means of the inverse image γ ∗E of the unique geometricmorphism γ E : E →
Set from E to Set ; the resulting internal category in E will be denoted by the symbol C .Indexed categories will be denoted by underlined letters, with possiblya subscript indicating the indexing category; the fibre at an object E of a E -indexed category A will be denoted by the symbol A E , and the functor A E → A E ′ corresponding to an arrow e : E → E ′ in E will be denoted bythe symbol A e .A E -indexed subcategory B E of a E -indexed category A E consists, foreach object every E ∈ E , of a subcategory B E of the category A E such thatfor any arrow α : E ′ → E in E the functor A α : A E → A E ′ restricts tothe subcategories B E and B E ′ . This clearly defines a E -indexed category B E with a E -indexed inclusion B E ֒ → A E .A E -indexed functor F : B E → A E is said to be full if for every object E of E the functor F E : A E → B E is full. A E -indexed subcategory B E of a E -indexed category A E is said to be a full E -indexed category of A E if theassociated E -indexed inclusion functor is full.Every Grothendieck topos E gives rise to a E -indexed category E E ob-tained by indexing E over itself.Recall that if C = ( d C , d C : C → C ) is an internal category in a topos E , a diagram of shape C in E is a pair ( f : F → C , φ : C × C F → F ) ofarrows in E satisfying appropriate conditions, where the pullback C × C F is taken relatively to the arrow d C : C → C and the arrow f : F → C : C × C F π F / / π (cid:15) (cid:15) F f (cid:15) (cid:15) C d C / / C . For a Grothendieck topos E and an internal category C in E , we have a E -indexed category [ C , E ] , whose underlying category is the category [ C , E ] of diagrams of shape C in E and morphisms between them.Any internal category C in E gives naturally rise to a E -indexed category,which we call the E -externalization of C and denote by the symbol C E . The7ategory [ C , E ] is equivalent (naturally in E ) to the category [ C E , E E ] E of E -indexed functors C E → E E and indexed natural transformations betweenthem (by Lemma B2.3.13 in [21]) and also to the category [ C , E ] (by CorollaryB2.3.14 in [21]). The equivalence between [ C , E ] and [ C , E ] restricts to anequivalence between the full subcategories Tors ( C , E ) of C -torsors in E (asin section B3.2 of [21]) and Flat ( C , E ) of flat functors C → E (as in chapterVII of [26]). For any internal functor between internal categories in E orinternal diagram F in E , we denote the corresponding E -indexed functor bythe symbol F E . For any functor F : C → E , we denote the internal diagramin [ C , E ] corresponding to it under the equivalence [ C , E ] ≃ [ C , E ] by thesymbol F .The discrete opfibration p : F → C over C corresponding to a diagram ( f : F → C , φ : C × C F → F ) of shape C in E is defined as follows: F = F , F = C × C F , d F = π F : F = C × C F → F = F , d F = φ : F = C × C F → F = F , p = f : F = F → C , p = π : F = C × C F → C .The discrete opfibration corresponding to a diagram F ∈ [ C , E ] will also bedenoted by π opfF : R opf F → C .For internal diagrams G ∈ [ C op , E ] it is also natural to consider the discrete fibration corresponding to G , i.e. the opposite functor π opfG op : R opf G op → C opop = C . We shall denote this functor by π fG : R f G → C .For any internal category C in a topos E , the category [ C , E ] of diagramsof shape C in E is equivalent to the category DOpf / C of discrete opfibrations over C (cf. Proposition B2.5.3 [21]); we shall denote this equivalence by τ C E : [ C , E ] → DOpf / C . A diagram of shape C in E lies in the subcategory Tors ( C , E ) of [ C , E ] ifand only if the domain of the corresponding discrete opfibration is a filteredinternal category in E .Any internal functor H : C → D between internal categories C and D in a topos E induces a functor [ D , E ] → [ C , E ] , denoted D → D ◦ H ,which corresponds, at the level of E -indexed categories, to the compositionfunctor with the indexed functor corresponding to H , and, at the level ofdiscrete opfibrations associated to the internal diagrams, as the pullback ofthem along the functor H . The latter pullbacks in the category of internalcategories in E are computed ‘pointwise’ as pullbacks in E , and they arepreserved by the dualizing functor C → C op .Let us recall from [29] the notion of (indexed) colimit of a E -indexedfunctor, where E is a cartesian category. We shall denote by ! I the uniquearrow from an object I of E to the terminal object of E . Given a S -indexedfunctor Γ : X → A and an object A of A , we denote by ∆ XA A the E -indexedfunctor X → A assigning to any E ∈ E the constant functor on X E withvalue A ! I ( A ) .Let X be any indexed category and Γ : X → A any indexed functor.8n indexed cocone µ : Γ → A consists of an object A in A together withan indexed natural transformation µ : Γ → ∆ XA A , i.e. for each I in E anordinary cocone µ I : Γ I → ∆( A ! I ( A )) such that for each arrow α : J → I in E , α ∗ ◦ µ I = µ J ◦ α ∗ . If µ : Γ → ∆ XA A is a universal such cone we say that it isa colimit cone over the indexed functor Γ with vertex A . If, furthermore, forany object I of E , the localization µ/I : Γ /I → (∆ XA A ) /I = ∆ X/IA/I ( A ! I ( A )) of µ at I is a colimit cone we say that µ is the indexed colimit of Γ /I .We can describe this notion more explicitly in the particular case of E -indexed functors with values in the E -indexed category E E . Let us supposethat A E is a E -indexed category and D : A E → E E is an indexed functor. Acocone µ over D consists of an object U of E and, for each object E of E ,of a cocone µ E : D E → ∆(! ∗ E ( U )) over the diagram D E : A E → E /E suchthat for any arrow α : E ′ → E in E we have α ∗ ( µ E ( c )) = µ E ′ ( A α ( c )) forall c ∈ A E , that is α ∗ µ E = µ E ′ A α as arrows D E ′ ( A α ( c )) = α ∗ ( D E ( c )) → ! E ′ ( U ) = α ∗ (! ∗ E ( U )) in E /E ′ , where α ∗ : E /E → E /E ′ is the pullback functor(notice that D E ′ ◦ A α = α ∗ ◦ D E since D is a E -indexed functor).Note that if A E is the E -externalization of an internal category C in E and D : A E → E E is an indexed functor corresponding to an internal diagram D ∈ [ C , E ] , the discrete opfibration associated to the internal diagram in [! ∗ E ( C ) , E /E ] corresponding to the localization D/E : A E /E → E E /E ∼ = E /E E /E is the image of the discrete opfibration associated to D under thepullback functor ! ∗ E : E → E /E along the unique arrow ! E : E → E .The colimit colim E ( D ) in E of an internal diagram D ∈ [ C , E ] , where C isan internal category in E , is defined to be the coequalizer of the two arrows d R opf D , d R opf D (recall that R opf D is the domain of the discrete opfibrationover C corresponding to the diagram D ).For any internal diagram G ∈ [ F , E ] , its colimit colim E ( G ) is isomorphicto the E -indexed colimit colim E ( G E ) of the E -indexed functor G E : F E → E E corresponding to it under the equivalence [ F , E ] ≃ [ F E , E E ] E .If D ∈ [ C , E ] is an internal diagram in E with colimit colim E ( D ) = coeq ( d R opf D , d R opf D ) , the colimiting E -indexed cocone ( colim E ( D ) , µ ) of thecorresponding E -indexed functor D E : C E → E E can be described as fol-lows. Let us denote by c the canonical coequalizer arrow ( R opf D ) → coeq ( d R opf D , d R opf D ) in E . For any object E of E and any object x : E → C of the category C E E , the arrow µ E ( x ) : D E ( x ) = r x → ! ∗ E ( colim E ( D )) = coeq ( d R opf D , d R opf D ) × E is equal to h c ◦ z x , r x i , where the arrow z x is de-fined by the following pullback diagram: R xr x (cid:15) (cid:15) z x / / ( R opf D ) π opfD ) (cid:15) (cid:15) E x / / C . .2 E -filtered indexed categories The following definition will be important in the sequel.
Definition 2.1.
Let E be a Grothendieck topos and A be a E -indexed cat-egory. We say that A is E -filtered if the following conditions are satisfied:(a) For any object E of E there exists an epimorphic family { e i : E i → E | i ∈ I } in E and for each i ∈ I an object b i of the category A E i ;(b) For any E ∈ E and any objects a and b of the category A E there exists anepimorphic family { e i : E i → E | i ∈ I } in E and for each i ∈ I an object c i of the category A E i and arrows f i : A e i ( a ) → c i and g i : A e i ( b ) → c i in the category A E i ;(c) For any object E of E and any two parallel arrows u, v : a → b in thecategory A E there exists an epimorphic family { e i : E i → E | i ∈ I } in E and for each i ∈ I an object c i of the category A E i and an arrow w i : A e i ( b ) → c i in A E i such that w i ◦ A e i ( u ) = w i ◦ A e i ( v ) .The E -externalization of any internal filtered category in E is E -filtered,but it is not true that if the externalization of an internal category C in E is E -filtered then C is filtered as an internal category in E .A standard example of indexed E -filtered categories is provided by in-dexed categories of elements of flat functors with values in E , in the sense ofthe following definition. Definition 2.2.
Let P : C op → E be a functor. The E - indexed category ofelements R P E of P assigns to any object E of E the category R P E whoseobjects are the pairs ( c, x ) where c is an object of C and x : E → F ( c ) is aan arrow in E , and whose arrows ( c, x ) → ( d, y ) are the arrows f : c → d in C such that F ( f ) ◦ y = x , and to any arrow e : E ′ → E in E the functor R P e : R P E → R P E ′ sending any object ( c, x ) of R P E to the object ( c, x ◦ e ) of R P E ′ and acting accordingly on the arrows. Proposition 2.3.
Let F : C op → E be a flat functor. Then the indexedcategory R F E is E -filtered. Proof
Straightforward from the characterization of flat functors as filteringfunctors given in chapter VII of [26]. (cid:3)
Given an internal diagram D ∈ [ C , E ] , the corresponding E -indexed functor D E : C E → E E can be described as follows. For any object E of E , D E : C E → E /E sends any object x : E → C of C E to the object r x : R x → E of E /E obtained by pulling ( π opfF ) : ( R opf F ) → C back along x , and any10rrow h : E → C of C E from x : E → C to x ′ : E → C to the arrow D E ( h ) : r x → r x ′ in E /E defined as follows. Consider the pullback squares S s (cid:15) (cid:15) u / / ( R opf D ) π opfD ) (cid:15) (cid:15) E h / / C and R x ′ r x ′ (cid:15) (cid:15) z x ′ / / ( R opf D ) π opfD ) (cid:15) (cid:15) E x ′ / / C . Since d C ◦ h = x ′ we have that ( π opfD ) ◦ d R opf D ◦ u = x ′ ◦ s and henceby the universal property of the first pullback square, there exists a uniquearrow β : S → R x ′ such that r x ′ ◦ β = s and z x ′ ◦ β = d R opf D ◦ u .As π opfF : R opf F → C is a discrete opfibration, the diagram ( R opf D ) π opfD ) (cid:15) (cid:15) d R opf D / / ( R opf D ) π opfD ) (cid:15) (cid:15) C d C / / C is a pullback. ‘Composing’ it with the first pullback square thus yields apullback square S s (cid:15) (cid:15) d R opf D ◦ u / / ( R opf D ) π opfD ) (cid:15) (cid:15) E d C ◦ h / / C . Now, since ( π opfD ) ◦ z x = x ◦ r x = d C ◦ h ◦ r x , the universal property ofthis pullback square provides a unique arrow γ : R x → S such that s ◦ γ = r x and d R opf D ◦ u ◦ γ = z x . We define D E ( h ) : r x → r x ′ in E /E to be equal tothe composite arrow β ◦ γ : R x → R x ′ . For any internal category F in E , we denote by coeq ( F ) the coequalizer ofthe two arrows d F and d F . 11rom the above discussion it follows that for any D, P ∈ [ C , E ] , colim E ( P ◦ π opfD ) ∼ = colim E ( D ◦ π opfP ) . Indeed, if we consider the pullback square R q D (cid:15) (cid:15) q P / / R opf P π opfP (cid:15) (cid:15) R opf D π opfD / / C in the category cat ( E ) of internal categories in E , we have that colim E ( P ◦ π opfD ) ∼ = coeq ( dom ( q D )) = coeq ( R ) = coeq ( dom ( q P )) ∼ = colim E ( D ◦ π opfP ) .Similarly, by exploiting the fact that the operation F → coeq ( F ) on in-ternal categories in E is invariant under the dualization functor F → F op , weobtain another commutation result, which we shall use in the sequel: for anyinternal diagram F ∈ [ C op , E ] and any internal diagram P ∈ [ C , E ] , we havea natural isomorphism colim E ( P ◦ π fF ) ∼ = colim E ( F ◦ π fP ) . To prove this, weconsider the following pullback squares: R q F (cid:15) (cid:15) q P / / R opf P π opfP (cid:15) (cid:15) S r P (cid:15) (cid:15) r F / / R opf F π opfF (cid:15) (cid:15) R f F π fF / / C R f P π fP / / C op . We have that colim E ( P ◦ π fF ) ∼ = coeq ( dom ( q F )) = coeq ( R ) , while colim E ( F ◦ π P f ) = coeq ( dom ( r P )) = coeq ( S ) . But the fact that the dualization functor preserves pullbacks in cat ( E ) im-plies that S ≃ R op , whence coeq ( R ) ∼ = coeq ( S ) , as required.Summarizing, we have the following Proposition 2.4.
Let C be an internal category in a topos E , P and D internal diagrams in [ C , E ] and F an internal diagram in [ C op , E ] . Then wehave natural isomorphisms(i) colim E ( P ◦ π opfD ) ∼ = colim E ( D ◦ π opfP ) ;(ii) colim E ( P ◦ π fF ) ∼ = colim E ( F ◦ π fP ) . (cid:3) F : C op → E , where C is a small category and E is a Grothendieck topos,and a functor P : C →
Set . To this end, we explicitly describe the discreteopfibration corresponding to the diagram of shape C in E , where C is theinternalization of C in E , associated to a functor G : C → E . We denote by Ob ( C ) the set of objects of C and by Arr ( C ) the set of arrows of C .We have that F = ` c ∈C G ( c ) and that f : F = ` c ∈C G ( c ) → C = ` c ∈C E isequal to ` c ∈C ! G ( c ) , where ! G ( c ) is the unique arrow G ( c ) → E in E (for any c ∈ C ).Let J f : G ( dom ( f )) → ` f ∈ Arr ( C ) G ( dom ( f )) (resp. µ c : G ( c ) → ` c ∈ Ob ( C ) G ( c ) , κ f : 1 E → ` f ∈ Arr ( C ) E , λ c : 1 E → ` c ∈ Ob ( C ) E ) be the canonical coproduct ar-rows.First, let us show that the diagram ` f ∈ Arr ( C ) G ( dom ( f )) d G / / ` f ∈ Arr ( C ) ! G ( dom ( f )) (cid:15) (cid:15) ` c ∈ Ob ( C ) G ( c ) ` c ∈ Ob ( C ) ! G ( c ) (cid:15) (cid:15) ` f ∈ Arr ( C ) E d C / / ` c ∈ Ob ( C ) E . is a pullback, where the arrow d G is defined by setting d G ◦ J f = µ dom ( f ) (forany f ∈ Arr ( C ) ). We have to prove that, for any object E of E and arrows α : E → ` f ∈ Arr ( C ) E and β : E → ` c ∈ Ob ( C ) G ( c ) such that ` c ∈ Ob ( C ) ! G ( c ) ◦ β = d C ◦ α , there exists a unique arrow γ : E → ` f ∈ Arr ( C ) ! G ( dom ( f )) such that α = ` f ∈ Arr ( C ) G ( dom ( f )) ◦ γ and β = d G ◦ γ . To this end, consider, for any c ∈ Ob ( C ) and f ∈ Arr ( C ) , the commutative diagram E ( f,c ) p c (cid:15) (cid:15) q f / / E fy f (cid:15) (cid:15) α f / / E κ f (cid:15) (cid:15) id " " ❋❋❋❋❋❋❋❋❋❋❋ E cβ c (cid:15) (cid:15) z c / / E α / / β (cid:15) (cid:15) ` f ∈ Arr ( C ) E d C (cid:15) (cid:15) E λ dom ( f ) } } ④④④④④④④④④④④ G ( c ) ! G ( c ) % % ❏❏❏❏❏❏❏❏❏❏❏❏ κ c / / ` c ∈ Ob ( C ) G ( c ) ` c ∈ Ob ( C ) ! G ( c ) / / ` c ∈ Ob ( C ) E E , λ c qqqqqqqqqqq ( f, c ) such that c = dom ( f ) , we have E ( f,c ) ∼ = 0 E . On the other hand,the stability of coproducts under pullbacks implies that E ∼ = ` c ∈ Ob ( C ) E c , and E ∼ = ` f ∈ Arr ( C ) E f ; whence E ∼ = ` ( f,c ) ∈ Arr ( C ) × Ob ( C ) E ( f,c ) ∼ = ` ( f,c ) | dom ( f )= c E ( f,c ) ,with canonical coproduct arrows ξ ( f,c ) = y f ◦ q f = z c ◦ p c : E ( f,c ) → E . Wedefine, for each pair ( f, c ) such that c = dom ( f ) , the arrow γ ( f,c ) : E ( f,c ) → ` f ∈ Arr ( C ) G ( dom ( f )) as the composite J f ◦ β c ◦ p c , and set γ equal to the ar-row ` ( f,c ) | dom ( f )= c γ ( f,c ) : E → ` f ∈ Arr ( C ) G ( dom ( f )) . We have to verify that α = ` f ∈ Arr ( C ) ! G ( dom ( f )) ◦ γ and β = d G ◦ γ or, equivalently, that for any pair ( f, c ) such that dom ( f ) = c , we have:(1) α ◦ ξ ( f,c ) = ` f ∈ Arr ( C ) ! G ( dom ( f )) ◦ γ ( f,c ) and(2) β ◦ ξ ( f,c ) = d G ◦ γ ( f,c ) .To prove (1) , we preliminarily show that for any pair ( f, c ) such that c = dom ( f ) , we have α f ◦ q f =! G ( c ) ◦ β c ◦ p c . Since the arrow λ c = λ dom ( f ) :1 E → ` c ∈ Ob ( C ) E is monic, it is equivalent to prove that λ dom ( f ) ◦ α f ◦ q f = λ c ◦ ! G ( c ) ◦ β c ◦ p c . But the commutativity of the above diagram yields λ dom ( f ) ◦ α f ◦ q f = d C ◦ κ f ◦ α f ◦ q f = d C ◦ α ◦ y f ◦ q f = ` c ∈ Ob ( C ) ! G ( c ) ◦ β ◦ y f ◦ q f = ` c ∈ Ob ( C ) ! G ( c ) ◦ β ◦ z c ◦ p c = ` c ∈ Ob ( C ) ! G ( c ) ◦ µ c ◦ β c ◦ p c = λ c ◦ ! G ( c ) ◦ β c ◦ p c , asrequired.We thus have α ◦ ξ ( f,c ) = α ◦ y f ◦ q f = κ f ◦ α f ◦ q f = κ f ◦ ! G ( c ) ◦ β c ◦ p c = ` f ∈ Arr ( C ) ! G ( dom ( f )) ◦ γ ( f,c ) . This proves condition (1) .Further, β ◦ ξ ( f,c ) = β ◦ z c ◦ p c = µ c ◦ β c ◦ p c = µ dom ( f ) ◦ β c ◦ p c = d G ◦ J f ◦ β c ◦ p c = d G ◦ γ ( f,c ) . This proves condition (2) .Let us define the arrow d G : a f ∈ Arr ( C ) G ( dom ( f )) → a c ∈ Ob ( C ) G ( c ) by setting, for each f ∈ Arr ( C ) , d G ◦ J f = µ cod ( f ) ◦ G ( f ) .The discrete opfibration p : F → C corresponding to G can be describedas follows: F = ` c ∈ Ob ( C ) G ( c ) , F = ` f ∈ Arr ( C ) G ( dom ( f )) , the domain andcodomain arrows d F , d F : F → F are respectively equal to d G and to14 G , p : F → C is equal to ` c ∈ Ob ( C ) ! G ( c ) and p : F → C is equal to ` f ∈ Arr ( C ) ! G ( dom ( f )) . The composition law in the internal category F is definedin the obvious way.We leave to the reader the straightforward task of verifying that thisis indeed the discrete opfibration corresponding to the functor G via thecomposite of the equivalence [ C , E ] ≃ [ C , E ] with the equivalence τ C E : [ C , E ] → DOpf / C . Recalling that, for any functor F : C op → E , the discrete fibration π fF : R f F → C associated to it is equal to π opfF op : R opf F op → C opop = C ,we deduce the following explicit description of the discrete fibration π fF : R f F → C associated to a functor F : C op → E : ( R f F ) = ` c ∈ Ob ( C ) F ( c ) , ( R f F ) = ` f ∈ Arr ( C ) F ( cod ( f )) , the domain and codomain arrows d F , d F :( R f F ) → ( R f F ) are defined by the conditions d F ◦ J f = µ cod ( f ) and d F ◦ J f = µ dom ( f ) ◦ F ( f ) for all f ∈ Arr ( C ) (where µ c : F ( c ) → ` c ∈ Ob ( C ) F ( c ) and J f : F ( cod ( f )) → ` f ∈ Arr ( C ) F ( cod ( f )) are the canonical coproduct arrows), ( π fF ) : ( R f F ) → C is equal to ` c ∈ Ob ( C ) ! F ( c ) and ( π fF ) : ( R f F ) → C is equal to ` f ∈ Arr ( C ) ! F ( cod ( f )) . The composition law in the internal category R f F is defined in the obvious way. Theorem 2.5.
Let F : C op → E be a functor from the opposite of a smallcategory C to a Grothendieck topos E and P : C →
Set be a functor. Thenthe following three objects are naturally isomorphic:(i) colim ( F ◦ π fP ) (ii) colim E ( F ◦ π fP E ) ∼ = colim E ( P E ◦ π fF ) (cf. Proposition 2.4)(iii) colim E ( P E E ◦ π fF E ) ,where P E is the internal diagram in [ C , E ] given by γ ∗E ◦ P . Proof
The isomorphism between colim E ( P E ◦ π fF ) and colim E ( P E E ◦ π fF E ) follows from the general fact that for any internal diagram G ∈ [ D , E ] itscolimit colim E ( G ) is isomorphic to the E -indexed colimit colim E ( G E ) . It thusremains to prove the isomorphism between colim ( F ◦ π fP ) and colim E ( F ◦ π fP E ) . To this end, we recall the following three general facts:151) For any functor G : D → E , its colimit colim ( G ) is naturally isomorphicto the colimit colim E ( G ) ;(2) For any functor H : D → C between small categories C and D and anyfunctor M : C → E , we have a natural isomorphism F ◦ H ∼ = F ◦ γ ∗E ( H ) ;(3) For any geometric morphism f : F → E , the diagram [ C , E ] f ∗ ( − ) (cid:15) (cid:15) τ C E / / DOpf / C f ∗ ( − ) (cid:15) (cid:15) [ f ∗ ( C ) , E ] τ f ∗ ( C ) F / / DOpf /f ∗ ( C ) commutes.We therefore have that colim ( F ◦ π fP ) ∼ = colim E ( F ◦ π fP ) ∼ = colim E ( F ◦ γ ∗E ( π fP )) ∼ = colim E ( F ◦ π fP E ) , where the first isomorphism follows from (1) , the second from (2) and thethird from (3) . (cid:3) We shall indicate the three isomorphic objects of the theorem by thesymbol F ⊗ C P .The following lemma is essentially contained in the proof of Giraud’stheorem (cf. [18] or, for instance, the Appendix of [26]), but we were notable to find its exact statement in the literature; we thus provide a proof ofit for the reader’s convenience. Lemma 2.6.
Let E be a Grothendieck topos and { e i : E i → E | i ∈ I } an epimorphic family in E . Then the arrow ` i ∈ I e i : ` i ∈ I E i → E yields anisomorphism ( ` i ∈ I E i ) /R ∼ = E , where R is the equivalence relation in E on theobject ` i ∈ I E i given by the subobject ` ( i,j ) ∈ I × I E i × E E j ` ( i,j ) ∈ I × I E i × E j ∼ = ` i ∈ I E i × ` j ∈ I E j ; in particular, the arrow ` i ∈ I e i : ` i ∈ I E i → E is the coequalizerin E of the two canonical arrows ` ( i,j ) ∈ I × I E i × E E j → ` i ∈ I E i . Proof
Let R be the kernel pair of the epimorphism ` i ∈ I e i , that is the pullbackof this arrow along itself; then, by the well-known exactness properties ofGrothendieck toposes, R is an equivalence relation on ` i ∈ I E i such that thecoequalizer in E of the two associated projections is isomorphic to q . Now,the fact that pullbacks preserve coproducts in a Grothendieck topos impliesthat R is isomorphic to the subobject ` ( i,j ) ∈ I × I E i × E E j ` ( i,j ) ∈ I × I E i × E j ∼ = ` i ∈ I E i × ` j ∈ I E j , as required. (cid:3) emark 2.7. The lemma admits an obvious generalization to arbitraryseparating sets for the topos E (cf. the Appendix of [26]). Corollary 2.8.
Let a : A → E and l : L → E be objects of the topos E /E ,and { e i : E i → E | i ∈ I } an epimorphic family in E . Then a family ofarrows { f i : e ∗ i ( a ) → e ∗ i ( l ) | i ∈ I } in the toposes E /E i defines a (unique)arrow f : a → l in E /E such that e ∗ i ( f ) = f i for all i ∈ I if and only if forevery i, j ∈ I , q ∗ i ( f i ) = q ∗ j ( f j ) , where the arrows q i and q j are defined by thefollowing pullback square: E i,jq j (cid:15) (cid:15) e i,j ! ! ❈❈❈❈❈❈❈❈ q i / / E ie i (cid:15) (cid:15) E j e j / / E . (cid:3) E -final subcategories The following definition will be important in the sequel. We shall borrowthe notation from section 2.4.
Definition 2.9.
Let A E be a E -indexed category and i : B E → A E be a E -indexed functor.(a) We say that i is E -final if for every E ∈ E and x ∈ A E there existsa non-empty set E xE of triplets { ( e i , b i , f i ) | i ∈ I } such that the family { e i : E i → E | i ∈ I } is epimorphic, b i is an object of B E i and f i : A e i ( x ) → i E i ( b i ) is an arrow in A E i (for each i ∈ I ) with the propertythat for any triplets { ( e i , b i , f i ) | i ∈ I } and { ( e ′ j , c j , f ′ j ) | j ∈ J } in E xE there exists an epimorphic family { g i,jk : E i,jk → E i,j | k ∈ K i,j } and foreach k ∈ K i,j an object d i,jk of B E i,jk and arrows r i,jk : B q i ◦ g i,jk ( b i ) → d i,jk and s i,jk : B q j ◦ g i,jk ( c j ) → d i,jk in B E i,jk such that i E i,jk ( r i,jk ) ◦ A q i ( f i ) = i E i,jk ( s i,jk ) ◦ A q j ( f ′ j ) .(b) We say that a E -indexed subcategory B E of a E -indexed category A E isa E -final subcategory of A E if the canonical E -indexed inclusion functor i : B E ֒ → A E is E -final, in other words if for every E ∈ E and x ∈ A E there exists a non-empty set E xE of triplets { ( e i , b i , f i ) | i ∈ I } such thatthe family { e i : E i → E | i ∈ I } is epimorphic, b i is an object of B E i and f i : A e i ( x ) → b i is an arrow in A E i (for each i ∈ I ) with the property thatfor any triplets { ( e i , b i , f i ) | i ∈ I } and { ( e ′ j , c j , f ′ j ) | j ∈ J } in E xE thereexists an epimorphic family { g i,jk : E i,jk → E i,j | k ∈ K i,j } and for each k ∈ K i,j an object d i,jk of B E i,jk and arrows r i,jk : B q i ◦ g i,jk ( b i ) → d i,jk and s i,jk : B q j ◦ g i,jk ( c j ) → d i,jk in B E i,jk such that r i,jk ◦ A q i ( f i ) = s i,jk ◦ A q j ( f ′ j ) .17c) We say that a E -indexed subcategory B E of a E -indexed category A E is a E -strictly final subcategory of A E if for any arrow f : a → b in A E thereexists an epimorphic family { e i : E i → E | i ∈ I } in E such that for any i ∈ I , the arrow A e i ( f ) : A e i ( a ) → A e i ( b ) lies in B E i .(d) We say that a E -indexed subcategory B E of a E -indexed category A E is E -full if for every arrow f : b → b ′ in A E , where b and b ′ are objects of B E , there exists an epimorphic family { e i : E i → E | i ∈ I } such thatfor any i ∈ I the arrow A e i ( f ) lies in B E i . Remarks 2.10. (a) Let B E be a E -indexed subcategory of A E with theproperty that for every object E of E and any object a ∈ A E α : E ′ → E ,there exists an epimorphic family { e i : E i → E | i ∈ I } in E such that forany i ∈ I , the object A e i ( a ) lies in B E i and for any arrow f : a → b in A E where a, b ∈ B E , there exists an epimorphic family { e i : E i → E | i ∈ I } in E such that for any i ∈ I , the arrow A e i ( f ) : A e i ( a ) → A e i ( b ) lies in B E i . Then B E is a E -strictly final subcategory of A E .(b) Every E -strictly final subcategory is a E -final subcategory.(c) If i is a E -full embedding of a E -indexed category B E into a E -filtered E -indexed category A E then i is E -final if and only if for every E ∈ E and x ∈ A E there exists a non-empty epimorphic family { e i : E i → E | i ∈ I } in E and for each i ∈ I an object b i of B E i and an arrow f i : A e i ( x ) → b i of A E i . Proposition 2.11.
Let B E be a E -full E -indexed subcategory of a E -indexedcategory A E . Then B E is E -filtered if and only if A E is E -filtered. Proof
The proof is entirely analogous to the classical one and left to thereader. (cid:3)
The following result represents a E -indexed version of the classical theo-rem formalizing the behaviour of colimits with respect to final subcategories. Theorem 2.12.
Let i : B E ֒ → A E be a E -final functor and D : A E → E E a E -indexed functor. Then D admits a colimit (resp. a E -indexed colimit) ifand only if D ◦ i admits a colimit (resp. a E -indexed colimit), and the twocolimits are equal. Proof
First, let us show that any cocone λ over D ◦ i with vertex V can be(uniquely) extended to a cocone ˜ λ over D . For any object a of A E , we haveto define an arrow ˜ λ E ( a ) : D E ( a ) → ! ∗ E ( V ) in E /E . By our hypotheses, thereexists an epimorphic family E = { e i : E i → E | i ∈ I } in E and a family ofarrows { f i : A e i ( a ) → i E i ( b i ) | i ∈ I } , where b i lies in B E i . Consider, for each i ∈ I , the arrow λ E i ( b i ) ◦ D E i ( f i ) : D E i ( A e i ( a )) = e ∗ i ( D E ( a )) → ! E i ( V ) = ∗ i (! ∗ E ( V )) . By applying the condition in the definition of E -final functor tothe pair (( e i , f i , b i ) , ( e i , f i , b i )) and exploiting the fact that λ is a cocone over D ◦ i , we obtain that q i ∗ ( λ E i ( b i ) ◦ D E i ( f i )) = q j ∗ ( λ E j ( b j ) ◦ D E j ( f j )) (herewe use the notation of Corollary 2.8). By Corollary 2.8, the family of arrows λ E i ( A e i ( a )) (for i ∈ I ) thus induces a unique arrow u E = D E ( a ) → ! ∗ E ( V ) .To be able to set ˜ λ E ( a ) equal to this arrow, we have to show that suchdefinition does not depend on the choice of the family { ( e i , f i ) } . But thisfollows similarly as above, by applying the condition in the definition of E -final functor and invoking the fact that λ is a cocone over D ◦ i .Notice that if a ∈ B E then ˜ λ E ( a ) = λ E ( a ) , and that for any cocone ξ over D , ξ = ˜ ξ ◦ i .Now that we have showed that for any a ∈ A E the definition of the arrow ˜ λ E ( a ) : D E ( a ) → ! ∗ E ( V ) in the topos E /E is well-posed, it remains to provethat the assignment a → ˜ λ E ( a ) defines a cocone over the diagram D E withvertex ! ∗ E ( V ) , i.e. that for any arrow f : a → b in A E , ˜ λ E ( b ) ◦ D E ( f ) = ˜ λ E ( a ) as arrows in E /E . Further, we have to show that the assignment E → ˜ λ E defines a E -indexed cone on the E -indexed functor D , i.e. that for any arrow α : E ′ → E in E and any a ∈ A E we have α ∗ (˜ λ E ( a )) = ˜ λ E ′ ( A α ( a )) . Thiscan be easily deduced from the fact that λ is an indexed cocone over D ◦ i .The second part of the theorem, for indexed colimits, follows from thefirst part by noticing that any localization of an indexed E -final functor withrespect to a localization functor E → E /E is a E /E -final functor. So it willbe sufficient to show the first part.To complete the proof of the theorem, it remains to show that for anycocone ( U, µ ) over D , ( U, µ ) is colimiting over D if and only if ( U, µ ◦ i ) iscolimiting over D .Suppose that D has a colimiting cocone µ with vertex U . We want toprove that µ ◦ i is a colimiting cocone for the functor D ◦ i . Let λ be a coconeover the diagram D ◦ i with vertex V ; then, as we have just proved, ˜ λ isa cocone over D with vertex V ; therefore ( U, µ ) factors through ( V, ˜ λ ) , byan arrow z : U → V in E . Clearly, z yields in particular a factorization of ( U, µ ◦ i ) across ( V, λ ) . The uniqueness of the factorization of ( U, µ ◦ i ) across ( V, λ ) follows from the fact that for any two cocones ξ and χ over D ◦ i , if ξ factors through χ by an arrow w then ˜ ξ factors through ˜ χ by the samearrow. This proves that if ( U, µ ) is a colimiting cocone over the diagram D then ( U, µ ◦ i ) is a colimiting cocone over the diagram D ◦ i .Conversely, suppose that ( U, µ ◦ i ) is a colimiting cocone over the diagram D ◦ i . ( Z, ˜ χ ) is a colimiting cocone over the diagram D . For any cocone ( Z, χ ) over D , we have that ( Z, χ ◦ i ) is a cocone over D ◦ i ; therefore ( U, µ ◦ i ) factorsuniquely through ( Z, χ ◦ i ) or equivalently (by similar arguments as above), ( Z, ˜ χ ) factors uniquely through ( U, µ ) . Therefore ( U, µ ) is a colimiting coconeover D , as required. (cid:3) Set E be the E -indexed category given by: Set E = Set for all E ∈ E and Set α = 1 Set (where Set is the identical functor on
Set ) for all arrows α in E . There is a E -indexed functor γ E : Set E → E E defined by: γ E E = γ ∗E /E : Set → E /E (for any object E of E ).We shall consider E -indexed functors obtained by composing the E -indexedfunctor γ E : Set E → E E with a E -indexed functor D : A E → Set E .Notice that a E -indexed functor D : A E → Set E consists of a functor D E : A E → Set for each object E of E such that for any arrow α : E ′ → E in E , D E ′ ◦ A α = D E .Let us give an explicit description of the cocones over the E -indexedfunctor γ E ◦ D . Specializing the general definition, we obtain that a cocone ( U, µ ) over γ E ◦ D consists of an object U of E and of an arrow µ E ( a ) : D E ( a ) → ! ∗ E ( U ) in E /E (for each objects E of E and a of A E ) such that forany arrow f : a → b in A E , µ E ′ ( b ) ◦ γ ∗E /E ( D E ( f )) = µ E ( a ) and for any arrow α : E ′ → E in E , α ∗ ( µ E ( a )) = µ E ′ ( A α ( a )) .By using the well-known adjunction between γ ∗E and the global sectionsfunctor E →
Set , we can alternatively present the above set of data asfollows: a cocone ( U, µ ) over γ E ◦ D consists of an object U of E and of afunction µ E ( a ) : D E ( a ) → Hom E ( E, U ) (for each objects E of E and a of A E ) such that for any arrow f : a → b in A E , µ E ′ ( b ) ◦ D E ( f ) = µ E ( a ) and forany arrow α : E ′ → E in E , Hom E ( α, U ) ◦ µ E ( a ) = µ E ′ ( A α ( a )) (notice thatthe domains of these two arrows are the same since D E ( a ) = D E ′ ( A α ( a )) ). Theorem 2.13.
Let F : C op → E be a functor. Then the E -indexed subcate-gory R F E of R F f E is strictly final. Proof
By definition of R F f E , for any object E of E the category R F f E hasas objects the arrows x : E → ` c ∈ Ob ( C ) F ( c ) and as arrows x → x ′ the arrows z : E → ` f ∈ Arr ( C ) F ( cod ( f )) such that d F ◦ z = x and d F ◦ z = x ′ . For any E ∈ E , the category R F E thus identifies as a subcategory of the category R F f E , through the assignment sending any pair ( c, x ) , where x : E → F ( c ) ,to the arrow µ c ◦ x : E → ` c ∈ Ob ( C ) F ( c ) and any arrow ( c, x ) → ( c ′ , x ′ ) tothe arrow J f ◦ x ′ : E → ` f ∈ Arr ( C ) F ( cod ( f )) (here we use the notations ofsection 2.4). It is clear that these assignments make R F E into a E -indexedsubcategory of R F f E .To prove our thesis, we shall apply the criterion of Remark 2.10(a). Toshow that the E -indexed subcategory R F E satisfies the first condition in the20emark, we observe that for any object x : E → ` c ∈ Ob ( C ) F ( c ) of the category R F f E , if we consider the pullbacks E ce c (cid:15) (cid:15) x c / / F ( c ) µ c (cid:15) (cid:15) E x / / ` c ∈ Ob ( C ) F ( c ) of x along the coproduct arrows µ c : F ( c ) → ` c ∈ Ob ( C ) F ( c ) , we obtain anepimorphic family { e c : E c → E | c ∈ Ob ( C ) } such that R F fe c ( x ) lies in thesubcategory R F E c .It remains to prove that the second condition of Remark 2.10(a) is sat-isfied. Let us suppose that ( c, x ) and ( c ′ , x ′ ) are objects of the subcategory R F E , and suppose that α : E → ` f ∈ Arr ( C ) F ( cod ( f )) is an arrow such that d F ◦ α = µ c ◦ x and d F ◦ α = µ c ′ ◦ x ′ : E x (cid:15) (cid:15) α / / ` f ∈ Arr ( C ) F ( cod ( f )) d F (cid:15) (cid:15) E x ′ (cid:15) (cid:15) α / / ` f ∈ Arr ( C ) F ( cod ( f )) d F (cid:15) (cid:15) F ( c ) µ c / / ` c ∈ Ob ( C ) F ( c ) F ( c ′ ) µ c ′ / / ` c ∈ Ob ( C ) F ( c ) . Consider, for each f ∈ Arr ( C ) , the following pullback square: E fe f (cid:15) (cid:15) α f / / F ( cod ( f )) J f (cid:15) (cid:15) E α / / ` f ∈ Arr ( C ) F ( cod ( f )) . The commutativity of the three diagrams above, combined with the factthat distinct coproduct arrows are disjoint, implies that for any arrow f of C such that dom ( f ) = c or cod ( f ) = c ′ we have E f ∼ = 0 E . We cantherefore restrict our attention to the arrows f such that dom ( f ) = c and cod ( f ) = c ′ . For any such arrow f , we have α f = x ′ ◦ e f (equivalently,since µ c ′ = µ cod ( f ) is monic, µ c ′ ◦ α f = µ c ′ ◦ x ′ ◦ e f ). Indeed, µ c ′ ◦ α f = d F ◦ J f ◦ α f = d F ◦ α ◦ e f = µ c ′ ◦ x ′ ◦ e f . Therefore α ◦ e f = J f ◦ x ′ ◦ e f ;indeed, α ◦ e f = J f ◦ α f = J f ◦ x ′ ◦ e f . Lastly, we observe that the arrow f defines an arrow ( c, x ◦ e f ) → ( c ′ , x ′ ◦ e f ) in the subcategory ( R F ) E f , i.e.21 ( f ) ◦ x ′ ◦ e f = x ◦ e f . Indeed, the arrow µ dom ( f ) = µ c ′ is monic and we have µ dom ( f ) ◦ F ( f ) ◦ x ′ ◦ e f = d F ◦ J f ◦ x ′ ◦ e f = J f ◦ α ◦ e f = µ c ◦ x ◦ e f .The epimorphic family { e f : E f → E | dom ( f ) = c and cod ( f ) = c ′ } thus satisfies the property that the arrow ( R F f E ) e f ( α ) lies in the subcategory ( R F ) E f . Our proof is therefore complete. (cid:3) Theorem 2.14.
Let C be a small category, E a Grothendieck topos, F : C op → E and P : C →
Set functors and P E the internal diagram in [ C , E ] given by γ ∗E ◦ P . Then the restriction of the functor ( P E E ◦ π fF E ) ∼ = P E ◦ π fF E to the E -indexed subcategory R F E of R F f E is naturally isomorphic to γ E ◦ z E ,where z E : R F E → Set E is the E -indexed functor defined by: z E E (( c, x )) = P ( c ) and z E E ( f ) = P ( f ) (for any E ∈ E , object ( c, x ) and arrow f in thecategory R F E ). Proof
We shall exhibit an isomorphism P E ◦ π fF E E (( c, x )) ∼ = ( γ E ◦ z E ) E (( c, x )) , natural in E ∈ E and ( c, x ) ∈ R F E .Consider the following pullback diagram in cat ( E ) : F ⊗ C P t F (cid:15) (cid:15) t P / / R opf P E π opfP E (cid:15) (cid:15) R f F π F / / C . Since coproducts are stable under pullback in a topos, we have that: ( F ⊗ C P ) = a c ∈ Ob ( C ) F ( c ) × γ ∗E ( P ( c )) , ( F ⊗ C P ) = a f ∈ Arr ( C ) F ( cod ( f )) × γ ∗E ( P ( dom ( f ))) ,t F : ( F ⊗ C P ) = a c ∈ Ob ( C ) F ( c ) × γ ∗E ( P ( c )) → ( Z f F ) = a c ∈ Ob ( C ) F ( c ) is equal to the arrow ` c ∈ Ob ( C ) π F ( c ) , where π F ( c ) is the canonical projection F ( c ) × γ ∗E ( P ( c )) → F ( c ) , t F : ( F ⊗ C P ) = a f ∈ Arr ( C ) F ( cod ( f )) × γ ∗E ( P ( dom ( f ))) → ( Z f F ) = a f ∈ Arr ( C ) F ( cod ( f ))
22s equal to the arrow ` f ∈ Arr ( C ) π ′ F ( cod ( f )) , where π ′ F ( cod ( f )) is the canonicalprojection F ( cod ( f )) × γ ∗E ( P ( dom ( f ))) → F ( cod ( f )) and the domain andcodomain arrows d F ⊗ C P , d F ⊗ C P : ( F ⊗ C P ) → ( F ⊗ C P ) are defined by the following conditions: d F ⊗ C P ◦ W f = Z cod ( f ) ◦ h π ′ F ( cod ( f )) , γ ∗E ( P ( f )) ◦ µ γ ∗E ( P ( dom ( f ))) i and d F ⊗ C P ◦ W f = Z dom ( f ) ◦ h F ( f ) ◦ π ′ F ( cod ( f )) , µ γ ∗E ( P ( dom ( f ))) i , where µ γ ∗E ( P ( dom ( f ))) is the canonical projection F ( cod ( f )) × γ ∗E ( P ( dom ( f ))) → γ ∗E ( P ( dom ( f ))) and W f : F ( cod ( f )) × γ ∗E ( P ( dom ( f ))) → a f ∈ Arr ( C ) F ( cod ( f )) × γ ∗E ( P ( dom ( f ))) ,Z c : F ( c ) × γ ∗E ( P ( c )) → a c ∈ Ob ( C ) F ( c ) × γ ∗E ( P ( c )) are the canonical coproduct arrows (respectively for f ∈ Arr ( C ) and c ∈ Ob ( C ) ).Now, the internal diagram P E ◦ π fF corresponds precisely to the discreteopfibration t F and hence the E -indexed functor P E ◦ π fF E sends any general-ized element x : E → ` c ∈ Ob ( C ) F ( c ) to the object of E /E given by the pullbackof t F along it. In particular, since in a topos any diagram of the form A if i (cid:15) (cid:15) / / ` i ∈ I A i ` i ∈ I f i (cid:15) (cid:15) B i / / ` i ∈ I B i , where the horizontal arrows are the canonical coproduct arrows, is a pull-back and products commute with pullbacks, the functor ( P E ◦ π fF ) E sendsany object ( c, x ) of the category R F E to the object given by the canonicalprojection E × γ ∗E ( P ( c )) → E . But this object is canonically isomorphic to γ ∗E /E ( P ( c )) in E /E , which is the value at ( c, x ) of the functor ( γ E ◦ z E ) E .Our proof is therefore complete. (cid:3) E -indexed colimiting cocone on therestriction to the E -indexed subcategory R F E of the E -indexed diagram P E ◦ π fF E (cf. section 2.1 for the background). The vertex of this colimitingcocone is the codomain of the coequalizer w : ( F ⊗ C P ) → colim E ( P E E ◦ π fF E ) of the pair of arrows d F ⊗ C P , d F ⊗ C P : ( F ⊗ C P ) → ( F ⊗ C P ) . For any ob-ject ( c, x ) of the category R F E , the colimit arrow ( P E ◦ π fF ) E )(( c, x )) ∼ = γ ∗E /E ( P ( c )) → colim E ( P E E ◦ π fF E ) is given by the composite ! ∗ E ( w ) ◦ h ( c,x ) ,where h ( c,x ) is the arrow ! ∗ E ( Z c ) ◦ h x × γ ∗E ( P ( c )) , π E i : γ ∗E /E ( P ( c )) ∼ = E × γ ∗E ( P ( c )) → ` c ∈ Ob ( C ) F ( c ) × γ ∗E ( P ( c )) × E ∼ =! ∗ E ( ` c ∈ Ob ( C ) F ( c ) × γ ∗E ( P ( c ))) ; inother words, the function ξ ( c,x ) : P ( c ) → Hom E ( E, colim E ( P E E ◦ π fF E )) corre-sponding to it assigns to every element a ∈ P ( c ) the arrow w ◦ Z c ◦ h x, r a ◦ ! E i ,where r a : 1 E → γ ∗E ( P ( c )) ∼ = ` a ∈ P ( c ) E is the coproduct arrow correspondingto the element a and ! E is the unique arrow E → E in E .It is immediate to verify that, under the isomorphism colim E ( P E E ◦ π fF E ) ∼ = colim ( F ◦ π fP ) of Theorem 2.5, the functions ξ ( c,x ) : P ( c ) → colim E ( P E E ◦ π fF E ) ∼ = colim ( F ◦ π fP ) admit the following description in terms of the colimiting arrows χ ( c,a ) : F ( c ) → colim ( F ◦ π fP ) : for any a ∈ P ( c ) , ξ ( c,x ) ( a ) = χ ( c,a ) ◦ x . E -indexed colimits In this section we establish necessary and sufficient conditions for a E -indexedcocone on a small diagram in E to be a ( E -indexed) colimit cone.Before stating and proving the main theorem of this section, we need tointroduce some relevant definitions and a few technical lemmas.Given a E -indexed functor D : A E → E E and an object R of the topos E , we denote by I DR the set of pairs of the form ( x, y ) , where x is an objectof the category A R and y is an arrow E /R → D R ( x ) in the topos E /R . Onsuch a set we consider the equivalence relation R DR generated by the pairsof the form (( x, y ) , ( x ′ , y ′ )) , where there exists an arrow f : x → x ′ in thecategory A R such that y ′ = D R ( f ) ◦ y .The following remarks are useful in connection with the application ofthe localization technique. Remarks 2.15. (a) For any E ∈ E , we have a natural bijection I DE ∼ = I D/E E /E , and the relation R DE corresponds to the relation R D/E E /E under thisbijection. 24b) For any arrow f : F → E and any object x ∈ A E , D F ( A f ( x )) = f ∗ ( D E ( x )) (this follows from the fact that D is a E -indexed functor).We shall denote the canonical arrow dom ( D F ( A f ( x ))) → dom ( D E ( x )) by the symbol r f .(c) For any object x of A E and any arrow y : ( f : F → E ) → D E ( x ) in thetopos E /E , there exists a unique arrow y f : (1 F : F → E ) → D F ( A f ( x )) in the topos E /F such that r f ◦ y f = y (where r f is considered here asan arrow in E /E in the obvious way);(d) For any E -indexed cocone µ over D with vertex U , any object x of A E and any arrow y : ( f : F → E ) → D E ( x ) in the topos E /E , we have µ E ( x ) ◦ y = (1 U × f ) ◦ µ F ( A f ( x )) ◦ y f . Moreover, we have pullbacksquares D F ( A f ( x )) µ F ( A f ( x )) (cid:15) (cid:15) r f / / D E ( x ) µ E ( x ) (cid:15) (cid:15) U × F π F (cid:15) (cid:15) U × f / / U × E π E (cid:15) (cid:15) F f / / E in E , where π E and π F are the obvious canonical projections. In par-ticular, for any other pair ( x ′ , y ′ ) , µ E ( x ) ◦ y = µ E ( x ′ ) ◦ y ′ if and only if µ F ( A f ( x )) ◦ y f = µ F ( A f ( x ′ )) ◦ y f ;(e) If A E is the E -externalization of an internal category C in E , x is ageneralized element E → C and y is an arrow ( f : F → E ) → D E ( x ) in E /E , there exists a unique arrow y f : (1 F : F → F ) → D F ( x ◦ f ) in E /F such that z x ◦ f ◦ y f = z x ◦ y (this arrow is provided by the universalproperty of the pullback square defining D F ( x ◦ f ) ), which coincides withthe arrow y f defined in Remark 2.15(c);(f) If a pair (( x, y ) , ( x ′ , y ′ )) belongs to R DR then for any indexed cocone λ over D with vertex V , λ R ( x ) ◦ y = λ R ( x ′ ) ◦ y ′ . Indeed, for any pair (( x, y ) , ( x ′ , y ′ )) with the property that there exists an arrow f : x → x ′ in the category A R such that y ′ = D E ( f ) ◦ y , we have that λ R ( x ′ ) ◦ y ′ = λ R ( x ′ ) ◦ D E ( f ) ◦ y = λ R ( x ) ◦ y (since λ R is a cocone). Lemma 2.16.
Let D ∈ [ C , E ] be an internal diagram, x : 1 → C and f : 1 → C generalized elements such that d C ◦ f = x and m : F → D ( x ) an arrow in E . Then the unique arrow χ m,f : F → ( R opf D ) in E such that d R opf D ◦ χ m,f = z x ◦ m and ( π opfD ) ◦ χ m,f = f ◦ ! F (provided by the universalproperty of the pullback square corresponding to the domains of the discreteopfibration associated to D ) satisfies the property that z x ◦ m = d R opf D ◦ χ m,f by definition) and z x ′ ◦ D ( f ) ◦ m = d R opf D ◦ χ m,f , where x ′ is equal to d C ◦ f and d R opf D , d R opf D are respectively the domain and codomain arrows ( R opf D ) → ( R opf D ) . Proof
By definition of D ( f ) , we have that z x ′ ◦ D ( f ) = d R opf D ◦ r ◦ ξ ,where r is defined by the pullback square H ! H (cid:15) (cid:15) r / / ( R opf D ) π opfD (cid:15) (cid:15) E f / / C and ξ is the unique arrow D ( x ) → H such that z x = d R opf D ◦ r ◦ ξ , providedby the universal property of the pullback square H s (cid:15) (cid:15) d R opf D ◦ r / / ( R opf D ) π opfD ) (cid:15) (cid:15) E d C ◦ f / / C . It thus remains to prove that χ m,f = r ◦ ξ ◦ m . But this immediatelyfollows from the universal property of the pullback R opf D π opfD (cid:15) (cid:15) d R opf D / / ( R opf D ) π opfD (cid:15) (cid:15) C d C / / C , since d R opf D ◦ χ m,f = d R opf D ◦ ( r ◦ ξ ◦ m ) and π opfD ◦ χ m,f = π opfD ◦ ( r ◦ ξ ◦ m ) . (cid:3) Lemma 2.17.
Let D be an internal diagram in [ C , E ] and χ be an arrow E → ( R opf D ) . Let x = ( π opfD ) ◦ d R opf D ◦ χ , x ′ = ( π opfD ) ◦ d R opf D ◦ χ and f = ( π opfD ) ◦ χ . Then d C ◦ f = x , d C ◦ f = x ′ and D ( f ) ◦ χ = χ , where χ is the unique arrow E → D ( x ) such that z x ◦ χ = d R opf D ◦ χ (whichexists by the universal property of the pullback square defining D ( x ) ) and χ is the unique arrow E → D ( x ′ ) such that z x ′ ◦ χ = d R opf D ◦ χ (whichexists by the universal property of the pullback square defining D ( x ′ ) . roof The first two identities follow straightforwardly from the definition.It remains to prove that D ( f ) ◦ χ = χ . Let us refer to the proof ofLemma 2.16 for notation. By the universal property of the pullback squaredefining D ( x ′ ) , it is equivalent to verify that z x ′ ◦ D ( f ) ◦ χ = z x ′ ◦ χ .But z x ′ ◦ D ( f ) ◦ χ = d R opf D ◦ r ◦ ξ ◦ χ , while z x ′ ◦ χ = d R opf D ◦ χ ; soto prove the desired equality it suffices to show that r ◦ ξ ◦ χ = χ . Thisfollows from the fact that d R opf D ◦ r ◦ ξ ◦ χ = z x ◦ χ = d R opf D ◦ χ by virtueof the universal property of the pullback square given by the domain of thediscrete opfibration associated to D . (cid:3) Proposition 2.18.
Let D ∈ [ C , E ] be an internal diagram, x, x ′ : 1 → C generalized elements and y : 1 E → D ( x ) , y ′ : 1 E → D ( x ′ ) arrows in E .Then ( z x ◦ y, z x ′ ◦ y ′ ) belongs to the equivalence relation on Hom E (1 E , ( R opf D ) ) generated by the pairs of the form ( d R opf D ◦ a, d R opf D ◦ a ) for some arrow a : 1 E → ( R opf D ) if and only if (( x, y ) , ( x ′ , y ′ )) belongs to R D E . Proof
Let us first prove the ‘if’ direction. We have that (( x, y ) , ( x ′ , y ′ )) belongs to R D E if and only if there exists a finite sequence ( x , y ) = ( x, y ) , . . . , ( x n , y n ) = ( x ′ , y ′ ) such that for any i ∈ { , . . . , n − } (1) either there exists f i : 1 → C such that d C ◦ f i = x i , d C ◦ f i = x i +1 and D ( f i ) ◦ y i = y i +1 or(2) there exists g i : 1 → C such that d C ◦ g i = x i +1 , d C ◦ g i = x i and D ( g i ) ◦ y i +1 = y i .In case (1) , Lemma 2.16 implies (by taking m to be y i and f to be f i ) that z x i ◦ y i = d R opf D ◦ χ y i ,f i and z x i +1 ◦ y i +1 = d R opf D ◦ χ y i ,f i (herethe notation is that of the lemma). In case (2) , Lemma 2.16 implies (bytaking m to be y i +1 and f to be g i ) that z x i +1 ◦ y i +1 = d R opf D ◦ χ y i +1 ,g i and z x i ◦ y i = d R opf D ◦ χ y i +1 ,g i .From this it clearly follows that ( z x ◦ y, z x ′ ◦ y ′ ) belongs to the equiva-lence relation T on Hom E (1 E , ( R opf D ) ) generated by the pairs of the form ( d R opf D ◦ a, d R opf D ◦ a ) for some arrow a : 1 E → ( R opf D ) , as required.Let us now prove the ‘only if’ direction. If ( z x ◦ y, z x ′ ◦ y ′ ) belongs to theequivalence relation T defined above, there exists a finite sequence χ , . . . , χ n of arrows E → ( R opf D ) and of arrows e i (which are either d R opf D or d R opf D - we denote by e op i the arrow d R opf D if e i is d R opf D and the arrow d R opf D e i is d R opf D ) such that e ◦ χ = z x ◦ y , e i +1 ◦ χ i +1 = e op i ◦ χ i for all i ∈ { , . . . , n − } and e n ◦ χ n = z x ′ ◦ y ′ .To deduce our thesis, it suffices to apply Lemma 2.17 noticing that if e i ◦ χ = z x ′′ ◦ y ′′ then χ = y ′′ if e i is d R opf D and χ = y ′′ if e i is d R opf D (where χ and χ are the arrows defined in the statement of Lemma 2.17)since z x ′′ ◦ χ = d R opf D ◦ χ = z x ′′ ◦ y ′′ (if e i is d R opf D ) and z x ′′ ◦ χ = d R opf D ◦ χ = z x ′′ ◦ y ′′ (if e i is d R opf D ). (cid:3) Theorem 2.19.
Let D : A E → E E be a E -indexed functor, where A E isequivalent to the E -externalization of an internal category in E . Then acocone µ over D with vertex U is an indexed colimiting cocone for D if andonly if the following conditions are satisfied:(i) For any object F of E and arrow h : F → U in the topos E , there existsan epimorphic family { f i : F i → F | i ∈ I } in E and for each i ∈ I anobject x i ∈ A F i and an arrow α i : 1 E /F i → D F i ( x i ) in the topos E /F i such that h h ◦ f i , F i i = µ F i ( x i ) ◦ α i as arrows E /F i → F ∗ i ( U ) in E /F i ;(ii) For any pairs ( x, y ) and ( x ′ , y ′ ) , where x and x ′ are objects of A E , y is an arrow ( f : F → E ) → D E ( x ) in E /E and y ′ is an arrow ( f : F → E ) → D E ( x ′ ) in E /E , we have µ E ( x ) ◦ y = µ E ( x ′ ) ◦ y ′ if andonly if there exists an epimorphic family { f i : F i → F | i ∈ I } in E such that the pair (( A f ◦ f i ( x ) , f ∗ i ( y )) , ( A f ◦ f i ( x ′ ) , f ∗ i ( y ′ )) belongs to therelation R DF i . Proof
First, let us prove that, under the assumption that A E is the E -externalization of an internal category C in E , the colimit cone for D satisfiesthe two conditions of the theorem.Under this assumption, condition ( i ) , applied to the colimiting cocone µ for D , rewrites as follows: for any object F of E and arrow h : F → colim ( D ) in E , there exists an epimorphic family { f i : F i → F | i ∈ I } in E and foreach i ∈ I a generalized element x i : F i → C and an arrow α i : 1 E /F i → D F i ( x i ) in the topos E /F i such that h h ◦ f i , F i i = c ◦ z x i ◦ α i as arrows ( f i : 1 E /F i → F ∗ i ( colim ( D )) in E /F i (where the notation is that of section2.1).Consider the following pullback square: F ′ h ′ (cid:15) (cid:15) f ′ / / F h (cid:15) (cid:15) ( R opf D ) c / / colim ( D ) . Since the arrow c is an epimorphism, the arrow f ′ : F ′ → F is an epimor-phism. This arrow will form, by itself, the single element of an epimorphic28amily satisfying condition ( i ) . We set x ′ equal to the composite ( π opfD ) ◦ h ′ (recall that ( π opfD ) : ( R opf D ) → C is the object component of the dis-crete opfibration associated to D ). Consider the following pullback squaredefining D F ′ ( x ′ ) : D F ′ ( x ′ ) v (cid:15) (cid:15) u / / ( R opf D ) π opfD ) (cid:15) (cid:15) F ′ x ′ / / C . By the universal property of this pullback square, there exists a unique arrow α x ′ : F ′ → D F ′ ( x ′ ) in E such that u ◦ α x ′ = h ′ and v ◦ α x ′ = 1 F ′ . So α x ′ is anarrow E /F ′ → D F ( x ′ ) in the topos E /F ′ , and h h ◦ f ′ , F ′ i = h c ◦ u ◦ α x ′ , v ◦ α x ′ i ,as required.Under the assumption that A E is the E -externalization of an internalcategory C in E , condition ( ii ) , applied to the colimiting cocone µ for D ,rewrites as follows: for any pairs ( x, y ) and ( x ′ , y ′ ) , where x and x ′ is ageneralized elements E → C , y is an arrow ( f : F → E ) → D E ( x ) in E /E and y ′ is an arrow ( f : F → E ) → D E ( x ′ ) in E /E , we have c ◦ z x ◦ y = c ◦ z x ′ ◦ y ′ if and only if there exists an epimorphic family { f i : F i → F | i ∈ I } in E such that the pair (( x ◦ f ◦ f i , y ◦ f i ) , ( x ′ ◦ f ◦ f i , y ′ ◦ f i )) belongs to the relation R DF i .Thanks to the localization technique, we can suppose E = 1 E withoutloss of generality. Indeed, condition ( ii ) for the diagram D , the cocone µ and the object E is equivalent to condition ( ii ) for the diagram D/E , thecocone µ/E and the object E /E , and µ is, by our assumption, an indexedcolimiting cocone and hence stable under localization.By Lemma 2.29, we have that c ◦ z x ◦ y = c ◦ z x ′ ◦ y ′ if and only ifthere exists an epimorphic family { f i : F i → F | i ∈ I } in E such thatfor each i ∈ I ( z x ◦ y ◦ f i , z x ′ ◦ y ′ ◦ f i ) belongs to the equivalence relationon the set Hom E ( F i , ( R opf D ) ) generated by the set of pairs of the form ( d R opf D ◦ a, d R opf D ◦ a ) , for a generalized element a : F i → ( R opf D ) in E .By Remark 2.15(e), we have that ( z x ◦ y ◦ f i , z x ′ ◦ y ′ ◦ f i ) = ( z x ◦ f ◦ f i ◦ ( y ◦ f i ) , z x ′ ◦ f ◦ f i ◦ ( y ′ ◦ f i )) . Our thesis then follows from Proposition 2.18, appliedto the toposes E /F i and the pairs (( x ◦ f ◦ f i , y ◦ f i ) , ( x ′ ◦ f ◦ f i , y ′ ◦ f i )) .Let us now prove that the conditions of the theorem are sufficient for µ to be a colimiting cocone. For this part of the theorem, we shall not need toassume the E -indexed category A E to be small.Since conditions ( i ) and ( ii ) are both stable under localization, it sufficesto prove that µ is a universal colimiting cocone over the diagram D . Supposethat λ is an indexed cocone over D with vertex V . We have to prove thatthere exists a unique arrow l : U → V in E such that for any E ∈ E andobject x of A E , ( l × E ) ◦ µ E ( x ) = λ E ( x ) . To define the arrow l we shall29efine a function L : Hom E ( E, U ) → Hom E ( E, V ) natural in E ∈ E . Given h ∈ Hom E ( E, U ) , by condition ( i ) in the statement of the theorem, thereexists an epimorphic family E = { e i : E i → E | i ∈ I } in E and for each i ∈ I an object x i of A E i and an arrow α i : 1 E /E i → D E i ( x i ) in E /E i such that h h ◦ e i , E i i = µ E i ( x i ) ◦ α i . The family of arrows { λ E i ( x i ) ◦ α i | i ∈ I } satisfiesthe hypotheses of Corollary 2.8. Indeed, using the notation of the corollary,we have that q ∗ i ( λ E i ( x i ) ◦ α i ) = q ∗ j ( λ E j ( x j ) ◦ α j ) for all i, j ∈ I ; this can beeasily proved by using condition ( ii ) of the theorem, the fact that the sameidentity holds for the cocone µ (in place of λ ) and Remark 2.15(d). Thereforethere exists a unique arrow L E h : E → V such that h L E h ◦ e i , E i i = λ E i ( x i ) ◦ α i for all i ∈ I . In order to be able to define L ( h ) as equal to L E h we need to checkthat for any other epimorphic family E ′ = { e ′ j : E ′ j → E | j ∈ J } in E , wehave L E h = L E ′ h . To this end, consider the fibered product of the epimorphicfamilies E and E ′ , that is the family of arrows p i,j = e i ◦ f i = e ′ j ◦ f ′ j : F i,j → E (for i, j ∈ I × J ), where the arrows f i and f ′ j are defined by the followingpullback square: F i,jf ′ j (cid:15) (cid:15) f i / / E ie i (cid:15) (cid:15) E j e ′ j / / E .
It clearly suffices to verify that for any i, j ∈ I , we have L E h ◦ p i,j = L E ′ h ◦ p i,j .Now, we have that h L E h ◦ e i , E i i = λ E i ( x i ) ◦ α i , while h L E ′ h ◦ e ′ j , E ′ j i = λ E ′ j ( x ′ j ) ◦ β j . Applying respectively f ∗ i and f ′∗ j to these two identities, weobtain that h L E h ◦ p i,j , F i,j i = λ F i,j ( x i,j ) ◦ f ∗ i ( α i ) and h L E ′ h ◦ p i,j , F i,j i = λ F i,j ( x ′ i,j ) ◦ f ′∗ j ( β j ) , where x i,j = A f i ( x i ) and x ′ i,j = A f ′ j ( x ′ j ) , from which itfollows that the condition L E h ◦ p i,j = L E ′ h ◦ p i,j is equivalent to the condition λ F i,j ( x i,j ) ◦ f ∗ i ( α i ) = λ F i,j ( x ′ i,j ) ◦ f ′∗ j ( β j ) . This condition can be proved byusing condition ( ii ) of the theorem, the fact that the same identity holdswith the cocone µ in place of λ and Remark 2.15(d). More specifically, theidentities h h ◦ e i , E i i = µ E i ( x i ) ◦ α i and h h ◦ e ′ j , E ′ j i = µ E ′ j ( x ′ j ) ◦ β i imply,applying respectively f ∗ i and f ′∗ j to them, the identity µ F i,j ( x i,j ) ◦ f ∗ i ( α i ) = µ F i,j ( x ′ i,j ) ◦ f ′∗ j ( β j ) . From condition ( ii ) of the theorem, it thus follows thatfor any i, j ∈ I × J , there exists an epimorphic family { g i,jk : G i,jk → F i,j | k ∈ K i,j } such that the pair (( A g i,jk ( x i,j ) , f ∗ i ( α i ) ◦ g i,jk ) , ( A g i,jk ( x ′ i,j ) , f ′∗ j ( β j ) ◦ g i,jk )) belongs to the relation R DG i,jk . This in turn implies (by Remark 2.15(d))that λ G i,jk ( A g i,jk ( x i,j )) ◦ ( f ∗ i ( α i ) ◦ g i,jk ) = λ G i,jk ( A g i,jk ( x ′ i,j )) ◦ ( f ′∗ j ( β j ) ◦ g i,jk ) .But λ G i,jk ( A g i,jk ( x i,j )) ◦ ( f ∗ i ( α i ) ◦ g i,jk ) = ( V × g i,jk ) ∗ ( λ F i,j ( x i,j ) ◦ f ∗ i ( α i )) and λ G i,jk ( A g i,jk ( x ′ i,j )) ◦ ( f ′∗ j ( β j ) ◦ g i,jk ) = ( V × g i,jk ) ∗ ( λ F i,j ( x ′ i,j ) ◦ f ′∗ j ( β j )) , whence ( V × g i,jk ) ∗ ( λ F i,j ( x i,j ) ◦ f ∗ i ( α i )) = ( V × g i,jk ) ∗ ( λ F i,j ( x ′ i,j ) ◦ f ′∗ j ( β j )) , equivalently30since the g i,jk are jointly epimorphic) λ F i,j ( x i,j ) ◦ f ∗ i ( α i ) = λ F i,j ( x ′ i,j ) ◦ f ′∗ j ( β j ) ,as required.It is clear that the assignment h → L ( h ) defined above is natural in E ∈ E ; it therefore remains to show that the resulting arrow l : U → V satisfies the required property that for any object E ∈ E and any object x of the category A E , ( l × E ) ◦ µ E ( x ) = λ E ( x ) . Take s to be the canonicalarrow s : D E ( x ) → E in E and set x ′ = A s ( x ) , E ′ = D E ( x ) ; then x ∈ A E ′ and, considered the pullback square D E ′ ( x ′ ) t (cid:15) (cid:15) r / / D E ( x ) s (cid:15) (cid:15) E ′ s / / E, the unique arrow α : E ′ → D E ′ ( x ′ ) such that r ◦ α = 1 E ′ and t ◦ α = 1 E ′ satisfies the property that h h, E ′ i = µ E ′ ( x ′ ) ◦ α . So the epimorphic family { E ′ : E ′ → E ′ } satisfies condition ( i ) of the theorem with respect to thearrow h = π U ◦ µ E ( x ) , where π U : U × E → U is the canonical projection,and hence l ◦ h = L ( h ) = λ E ′ ( x ′ ) ◦ α ; therefore, ( l × E ) ◦ µ E ( x ) = λ E ( x ) , asrequired.The proof of the theorem is now complete. (cid:3) Remark 2.20. (a) The proof of the theorem shows that the sufficiency ofthe conditions of the theorem holds more in general for any (i.e., notnecessarily small) E -indexed category A E ;(b) The conditions in the statement of the theorem are both stable underlocalization; that is, if the cone µ over the diagram D satisfies them thenfor any E ∈ E the cone µ/E over the diagram D/E does.
Proposition 2.21.
Let B E be a E -final subcategory of an indexed category A E , D : A E → E E a E -indexed functor and µ a E -indexed cocone over D .Then µ satisfies the conditions of Theorem 2.19 with respect to the diagram D if and only if µ ◦ i satisfies them with respect to the diagram D ◦ i , where i is the canonical embedding B E ֒ → A E . Proof
Suppose that µ satisfies the conditions of Theorem 2.19 with respectto the diagram D . The fact that µ ◦ i satisfies condition ( i ) of the theoremwith respect to the diagram D ◦ i immediately follows from the fact that µ does with respect to the diagram D , by using the fact that for each object x i ∈ A F i there exists an epimorphic family { g ik : G ik → F i | k ∈ K i } such that A g ik ( x i ) lies in B G ik and for any arrow α i : 1 E /F i → D F i ( x i ) in the topos E /F i ,its image under the pullback functor g ik ∗ is an arrow E /G ik → D G ik ( A g ik ( x i )) in the topos E /G ik . Let us now show that µ ◦ i satisfies condition ( ii ) withrespect to the diagram D ◦ i . 31et us first establish the following fact ( ∗ ) : for any pairs ( x, y ) and ( x ′ , y ′ ) in I DR (see the beginning of this section for the notation), if (( x, y ) , ( x ′ , y ′ )) ∈R DR , there exists an epimorphic family { r i : R i → R | i ∈ I } in E such that (( A r i ( x ) , r ∗ i ( y )) , ( A r i ( x ′ ) , r ∗ i ( y ′ ))) ∈ R D ◦ iR i .If (( x, y ) , ( x ′ , y ′ )) ∈ R DR , there exists a finite sequence ( x, y ) = ( x , y ) , . . . , ( x n , y n ) = ( x ′ , y ′ ) of pairs in I DR such that for any j ∈ { , . . . , n − } , either there exists anarrow f j : x j → x j +1 in A R such that y j +1 = D R ( f j ) ◦ y j or there exists anarrow f j : x j +1 → x j in A R such that y j = D R ( f j ) ◦ y j +1 . By applying thedefinition of E -final subcategory a finite number of times (using the fact thatthe fibered product of epimorphic families is again an epimorphic family),we can find an epimorphic family { r i : R i → R | i ∈ I } in E such that forevery i ∈ I and any j ∈ { , . . . , n − } , the arrow A r i ( f j ) lies in B R i . Fromthis it immediately follows that (( A r i ( x ) , r ∗ i ( y )) , ( A r i ( x ′ ) , r ∗ i ( y ′ ))) ∈ R D ◦ iR i , asrequired.Using fact ( ∗ ) , the proof of the fact that µ ◦ i satisfies condition ( ii ) ofthe theorem with respect to the diagram D ◦ i follows straightforwardly fromthe fact that µ does with respect to the diagram D .Conversely, let us suppose that µ ◦ i satisfies the conditions of the theoremwith respect to D ◦ i and deduce that µ does with respect to D . The fact thatthe validity of condition ( i ) for µ ◦ i with respect to D ◦ i implies the validityof condition ( i ) for µ with respect to D is obvious. Concerning condition ( ii ) for ( µ, D ) , this can be deduced from condition ( ii ) for ( µ ◦ i, D ◦ i ) by usingthe fact that, given ( x, y ) and ( x ′ , y ′ ) as in the statement of the condition for ( µ, D ) , there exists an epimorphic family { e j : E j → E | j ∈ J } such that A e j ( x ) and A e j ( x ′ ) lie in B E j for all j ∈ J .The proof of the proposition is now complete. (cid:3) Remark 2.22.
The proposition shows in particular that (the necessity ofthe conditions of) Theorem 2.19 not only holds for E -indexed categories A E which are E -small (i.e., which are equivalent to the E -externalization of aninternal category in E .), but for any E -indexed category which is a E -finalsubcategory of a E -small E -indexed category (for instance, to the E -indexedcategories of the form R F E , for a functor F : C op → E - cf. Theorem 2.13). Proposition 2.23.
Let D : A E → E E an indexed functor, where A E is a E -filtered category, R an object of E , ( x, y ) , ( x ′ , y ′ ) pairs in I DR . Then thereexists an epimorphic family { r i : R i → R | i ∈ I } in E such that for all i ∈ I , (( A r i ( x ) , r ∗ i ( y )) , ( A r i ( x ′ ) , r ∗ i ( y ′ ))) ∈ R DR i if and only if there exists anepimorphic family { e j : E j → R | j ∈ J } in E with the property that for any j ∈ J there exist arrows f j : A e j ( x ) → z and g j : A e j ( x ′ ) → z in the category A E j such that D E j ( f j ) ◦ y = D E j ( g j ) ◦ y ′ . roof The ‘if’ direction is obvious, so it remains to prove the ‘only if’one. It clearly suffices to prove, by induction on the length of a sequence ( x , y ) , . . . , ( x n , y n ) of pairs in I DR with the property that for any i ∈{ , . . . , n − } , either there exists an arrow f j : x j → x j +1 in A R suchthat y j +1 = D R ( f j ) ◦ y j or there exists an arrow f j : x j +1 → x j in A R suchthat y j = D R ( f j ) ◦ y j +1 , that there exists an epimorphic family { r i : R i → R | i ∈ I } in E such that for all i ∈ I , (( A r i ( x ) , r ∗ i ( y )) , ( A r i ( x n ) , r ∗ i ( y n ))) ∈R DR i . The case n = 1 is obvious. Suppose now that the claim is valid for n and prove that it holds for n + 1 . There exists an epimorphic family { e j : E j → R | j ∈ J } in E such that for any j ∈ J there exist arrows f j : A e j ( x ) → z and g j : A e j ( x n ) → z in the category A E j such that D E j ( f j ) ◦ y n = D E j ( g j ) ◦ y ′ n . On the other hand, there exists either an arrow g : x n → x n +1 in A R such that D R ( g ) ◦ y n = y n +1 or an arrow h : x n +1 → x n in A R such that D R ( g ) ◦ y n +1 = y n . In the latter case, the thesis followsstraightforwardly; so we can concentrate on the case when there exists anarrow g : x n → x n +1 in A R such that D R ( g ) ◦ y n = y n +1 .By using the definition of E -final subcategory, we obtain an epimorphicfamily { e j : E j → R | j ∈ J } in E and for each j ∈ J two arrows f j : A e j ( x ) → z and g j : A e j ( x n ) → z such that D E j ( f j ) ◦ e ∗ j ( y ) = D E j ( g j ) ◦ e ∗ j ( y n ) . Similarly, we obtain, for each j ∈ J , an epimorphic family { f jk : F jk → E j | k ∈ K j } in E and for each k ∈ K j arrows m jk : A f jk ( z ) → w jk and n jk : A f jk ( A e j ( x n +1 )) → w jk such that D F jk ( m jk ) ◦ D F jk ( A f jk ( g j )) ◦ ( f jk ◦ e j ) ∗ ( y n ) = D F jk ( n jk ) ◦ D F jk ( A f jk ( g )) ◦ ( f jk ◦ e j ) ∗ ( y n +1 ) . Now, by definition of E -final subcategory, for each j ∈ J and k ∈ K j there exists an epimorphicfamily { g k,jl : G k,jl → F jk | l ∈ L k,j } and for each l ∈ L k,j an arrow p k,ll : A g k,jl ( w jk ) → v k,jl such that p k,ll ◦ m jk ◦ A f jk ( g j ) = p k,jl ◦ n jk ◦ A f jk ( A e j ( g )) .For any j ∈ J, k ∈ K j and l ∈ L k,j , set a j,k,l : A e j ◦ f jk ◦ g k,jl ( x ) → v k,jl equalto p k,jl ◦ A g j,kl ( m jk ) ◦ A g k,jl ◦ f jk ( f j ) and b j,k,l : A e j ◦ f jk ◦ g k,jl ( x n ) → v k,jl equal to p k,jl ◦ A g j,kl ( n jk ) ◦ A g k,jl ◦ f jk ◦ e j ( g ) . It is readily seen that D G k,jl ( a j,k,l ) ◦ ( e j ◦ f jk ◦ g k,jl ) ∗ ( y ) = D G k,jl ( b j,k,l ) ◦ ( e j ◦ f jk ◦ g k,jl ) ∗ ( y n +1 ) , whence the requiredcondition is satisfied (by taking as epimorphic family { e j ◦ f jk ◦ g k,jl | j ∈ J, k ∈ K j , l ∈ L k,j } ). (cid:3) Combining the proposition with Theorem 2.19, we immediately obtainthe following result.
Corollary 2.24.
Let D : A E → E E an indexed functor, where A E is a E -filtered E -final subcategory of a small E -indexed category. Then a cocone µ over D with vertex U is a indexed colimiting cocone for D if and only if thefollowing conditions are satisfied: i) For any object E of E and arrow h : F → U in the topos E , there existsan epimorphic family { f i : F i → F | i ∈ I } in E ) and for each i ∈ I anobject x i ∈ A F i and an arrow α i : 1 E /F i → D F i ( x i ) in the topos E /F i such that h h ◦ f i , F i i = µ F i ( x i ) ◦ α i as arrows E /F i → F ∗ i ( U ) in E /F i ;(ii) For any pairs ( x, y ) and ( x ′ , y ′ ) , where x and x ′ are objects of A E , y is an arrow ( f : F → E ) → D E ( x ) in E /E and y ′ is an arrow ( f : F → E ) → D E ( x ′ ) in E /E , we have that µ E ( x ) ◦ y = µ E ( x ′ ) ◦ y ′ ifand only if there exists an epimorphic family { f i : F i → F | i ∈ I } in E and for each i ∈ I arrows g i : A f ◦ f i ( x ) → z i and h i : A f ◦ f i ( x ′ ) → z i in the category A F i such that D F i ( g i ) ◦ f ∗ i ( y ) = D F i ( h i ) ◦ f ∗ i ( y ′ ) .Moreover, if µ is colimiting for D then the following ‘joint embeddingproperty’ holds: for any pairs ( x, y ) and ( x ′ , y ′ ) , where x and x ′ are objectsof A E , y is an arrow ( f : F → E ) → D E ( x ) in E /E and y ′ is an arrow ( f : F → E ) → D E ( x ′ ) in E /E , there exists an epimorphic family { e i : F i → F | i ∈ I } in E and for each i ∈ I arrows g i : A f ◦ e i ( x ) → z i and h i : A f ◦ e i ( x ′ ) → z i in the category A F i such that D F i ( g i ) ◦ f ∗ i ( y ) = D F i ( h i ) ◦ f ∗ i ( y ′ ) and the diagram D F i ( A f ◦ e i ( x )) D Fi ( g i ) ' ' ❖❖❖❖❖❖❖❖❖❖❖ ξ ( Fi, A f ◦ ei ( x )) , , ❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳❳ D F i ( z i ) ξ ( Ei,zi ) / / colim ( D ) × F i D F i ( A f ◦ e i ( x ′ )) D Fi ( h i ) ♦♦♦♦♦♦♦♦♦♦♦ ξ ( Fi, A f ◦ ei ( x ′ )) ❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢❢ , where ξ ( E,x ) : D E ( x ) → colim ( D ) × E are the colimit arrows, commutes. Remark 2.25. (a) In the statement of the corollary, we can suppose with-out loss of generality the object f : F → E of the topos E /E to be equalto the terminal E : E → E . Indeed, by Remark 2.15, for any object x of A E and any arrow y : ( f : F → E ) → D E ( x ) in the topos E /E , µ E ( x ) ◦ y = µ E ( x ′ ) ◦ y ′ if and only if µ F ( A f ( x )) ◦ y f = µ F ( A f ( x ′ )) ◦ y f .(b) For any pairs ( x, y ) and ( x ′ , y ′ ) , where x and x ′ are objects of A E , y isan arrow ( f : F → E ) → D E ( x ) in E /E , y ′ is an arrow ( f : F → E ) → D E ( x ′ ) in E /E , and any epimorphic family { g j : G j → F | j ∈ J } in E ,condition ( ii ) of Corollary 2.24 is satisfied by the pair (( x, y ) , ( x ′ , y ′ )) ifand only if it is satisfied by the pair (( x, y ◦ g j ) , ( x ′ , y ′ ◦ g j )) for all j ∈ J .Let us now apply Corollary 2.24 in the context of a flat functor F : C op → E and a functor P : C →
Set , where C is a small category and E is a Grothendieck topos. Consider the restriction of the functor ( P E E ◦ π fF E ) ∼ = P E ◦ π fF E (where P E is the internal diagram [ C , E ] given by γ ∗E ◦ P )34o the E -indexed subcategory R F E of R F f E . Recall that, by Theorem 2.14,the E -indexed functor ( P E E ◦ π fF E ) is naturally isomorphic to the compositefunctor γ E ◦ z E , where z E : R F E → Set E is the E -indexed functor definedby: z E E (( c, x )) = P ( c ) and z E E ( f ) = P ( f ) (for any E ∈ E , object ( c, x ) andarrow f in the category R F E ).As observed in section 2.5, a E -indexed cocone µ over this diagramwith vertex U can be identified with a family of functions µ ( c,x ) : P ( c ) → Hom E ( E, U ) indexed by the pairs ( c, x ) consisting of an object c of C and ageneralized element x : E → F ( c ) satisfying the following properties:(i) For any generalized element x : E → F ( c ) and any arrow f : d → c in C , µ ( c,x ) ◦ P ( f ) = µ ( c,F ( f ) ◦ x ) ;(ii) For any generalized element x : E → F ( c ) and any arrow e : E ′ → E , µ ( c,x ◦ e ) = Hom E ( e, U ) ◦ µ ( c,x ) .The following theorem characterizes the cocones µ which are colimiting. Theorem 2.26.
Let C be a small category, E a Grothendieck topos, P : C →
Set a functor and F : C op → E a flat functor. Then a E -indexed cocone µ = { µ ( c,x ) : P ( c ) → Hom E ( E, U ) | c ∈ C , x : E → F ( c ) in E} with vertex U over the diagram given by the restriction of the functor ( P E E ◦ π fF E ) ∼ = P E ◦ π fF E (where P E is the internal diagram [ C , E ] given by γ ∗E ◦ P ) to the E -indexed subcategory R F E is colimiting if and only if the following conditionsare satisfied:(i) For any generalized element x : E → U there exists an epimorphicfamily { e i : E i → E | i ∈ I } in E , for each index i ∈ I an object c i in C , a generalized element x i : E i → F ( c i ) and an element y i ∈ P ( c i ) such that µ ( c i ,x i ) ( y i ) = x ◦ e i ;(ii) For any pairs ( a, x ) and ( b, x ′ ) , where a and b are objects of C and x : E → F ( a ) , x ′ : E → F ( b ) are generalized elements, and anyelements y ∈ P ( a ) and y ′ ∈ P ( b ) , µ ( a,x ) ( y ) = µ ( b,x ′ ) ( y ′ ) if and onlyif there exists an epimorphic family { e i : E i → E | i ∈ I } in E in E and for each index i ∈ I an object c i of C , arrows f i : a → c i , g i : b → c i in C and a generalized element x i : E i → F ( c i ) such that h x, x ′ i ◦ e i = h F ( f i ) , F ( g i ) i ◦ x i for all i ∈ I and P ( f i )( y ) = P ( g i )( y ′ ) .Moreover, if µ is colimiting the following ‘joint embedding property’ holds:for any pairs ( a, x ) and ( b, x ′ ) , where a and b are objects of C and x : E → F ( a ) , x ′ : E → F ( b ) are generalized elements, there exists an epimorphicfamily { e i : E i → E | i ∈ I } in E and for each index i ∈ I an object c i of C ,arrows f i : a → c i , g i : b → c i in C and a generalized element x i : E i → F ( c i ) uch that h x, x ′ i ◦ e i = h F ( f i ) , F ( g i ) i ◦ x i for all i ∈ I and (by Lemma 3.10)the following diagram commutes: P ( a ) P ( f ) ●●●●●●●● µ ( a,x ) / / Hom E ( E, U ) Hom E ( e i ,M ) ( ( ◗◗◗◗◗◗◗◗◗◗◗◗ P ( c i ) µ ( ci,xi ) / / Hom E ( E i , U ) P ( b ) P ( g ) ; ; ✇✇✇✇✇✇✇✇ µ ( b,x ′ ) / / Hom E ( E, U ) Hom E ( e i ,U ) ♠♠♠♠♠♠♠♠♠♠♠♠ In fact, the epimorphic family { e i : E i → E | i ∈ I } can be taken tobe the pullback of the family of arrows h F ( f ) , F ( g ) i : F ( c ) → F ( a ) × F ( b ) (for all spans ( f : a → c, g : b → c ) in the category C ) along the arrow h x, x ′ i : E → F ( a ) × F ( b ) . Proof
The theorem can be deduced from Corollary 2.24 by reasoning asfollows. Concerning condition ( i ) , its equivalence with condition ( i ) of Corol-lary 2.24 follows from Remark 2.25(a) and the fact that for any object ( z, c ) of the category R F E and any arrow α : 1 E /E → D E (( z, c )) in the topos E /E , denoting by ! f r : f r : F r → E → E /E the pullback of the coproductarrow u r : 1 E /E → γ ∗E /E ( P ( c )) ∼ = D E (( z, c )) in E /E (for each r ∈ P ( c ) ), thearrows ! f r define, for r ∈ P ( c ) , an epimorphic family in E /E on the object E /E ; indeed, this shows that we can suppose without loss of generality α tofactor through one of the coproduct arrows u r and hence to correspond toan element y ∈ P ( c ) .Concerning condition ( ii ) , by Remark 2.25 we can suppose without lossof generality that the pairs ( x, y ) and ( x ′ , y ′ ) , where x (resp. x ′ ) is an objectof R F E , y is an arrow ( f : F → E ) → D E ( x ) in E /E and y ′ is an arrow ( f : F → E ) → D E ( x ′ ) in E /E , have respectively the form (( x, c ) , y ) ,where y = u r ◦ ! f r : F r → E and (( x ′ , c ′ ) , y ′ ) , where y ′ = u r ′ ◦ ! f r ′ (with the abovenotation), for some r, r ′ ∈ P ( c ) . Thus we have that µ E ( x ) ◦ y = µ E ( x ′ ) ◦ y ′ if and only if ( Hom E ( f r , U ) ◦ µ ( x,c ) )( r ) = ( Hom E ( f r ′ , U ) ◦ µ ( x ′ ,c ′ ) )( r ′ ) ; but ( Hom E ( f r , U ) ◦ µ ( x,c ) )( r ) = µ ( c,x ◦ f r ) ( r ) and ( Hom E ( f r ′ , U ) ◦ µ ( x ′ ,c ′ ) )( r ′ ) = µ ( c ′ ,x ′ ◦ f r ′ ) ( r ′ ) , whence ( Hom E ( f r , U ) ◦ µ ( x,c ) )( r ) = ( Hom E ( f r ′ , U ) ◦ µ ( x ′ ,c ′ ) )( r ′ ) if and only if µ ( c,x ◦ f r ) ( r ) = µ ( c ′ ,x ◦ f r ′ ) ( r ′ ) . (cid:3) Recall from [26] (cf. p. 355) that the colimit of a functor H : I → E withvalues in a Grothendieck topos E can always be realized as the coequalizer q : ` i ∈ I H ( i ) → colim ( H ) of the pair of arrows a, b : a u : i → j in I H ( i ) → a i ∈ I H ( i ) a ◦ λ u = κ i and b ◦ λ u = κ j ◦ H ( u ) (for every arrow u : i → j in I ), where λ u : H ( dom ( u )) → ` u : i → j in I H ( i ) (forany u : i → j in I ) and κ i : H ( i ) → ` i ∈ I H ( i ) (for any i ∈ I ) are the canonicalcoproduct arrows.Given two arrows s, t : E → ` i ∈ I H ( i ) , for every objects i, j ∈ I , we canconsider the following pullback diagrams: E sip si (cid:15) (cid:15) a si / / H ( i ) κ i (cid:15) (cid:15) E tjp tj (cid:15) (cid:15) a tj / / H ( j ) κ j (cid:15) (cid:15) E s / / ` i ∈ I H ( i ) E t / / ` i ∈ I H ( i ) Let us also consider, for each i, j ∈ J , the pullback square E s,ti,jq ti,j (cid:15) (cid:15) q si,j / / E sip si (cid:15) (cid:15) E tj p tj / / E and denote by r s,ti,j : E s,ti,j → E the arrow p si ◦ q si,j = p tj ◦ q ti,j . Notice that thefamily of arrows { r s,ti,j | i, j ∈ I} is epimorphic.We shall use these notations throughout the section.The following lemma provides a characterization of the coequalizer of thefunctor H , which will be useful to us later on. Lemma 2.27.
Let p : ` i ∈ I H ( i ) → A be an epimorphism. Then p is iso-morphic to the canonical map q : ` i ∈ I H ( i ) → colim ( H ) if and only if for anyobject E of E , any objects i, j ∈ I and any generalized elements z : E → H ( i ) and w : E → H ( j ) , ( κ i ◦ z, κ j ◦ w ) ∈ R E if and only if p ◦ z = p ◦ w . Proof
As in a topos every epimorphism is the coequalizer of its kernel pair,any two epimorphisms α : B → C and β : B → D in E are isomorphic in B/ E if and only if their kernel pairs are isomorphic as subobjects of B × B . Inparticular, p is isomorphic to q if and only if the kernel pair of p is isomorphicto R . 37he kernel pair S of p satisfies the property that for any generalizedelements s, t : E → ` i ∈ I H ( i ) , ( s, t ) ∈ S E if and only if p ◦ s = p ◦ t . It thusremains to prove that the following two conditions are equivalent:(1) for any generalized elements s, t : E → ` i ∈ I H ( i ) , p ◦ s = p ◦ t if and onlyif ( s, t ) ∈ R E ;(2) for any object E of E , any objects i, j ∈ I and any generalized elements z : E → H ( i ) and w : E → H ( j ) , p ◦ κ i ◦ z = p ◦ κ j ◦ w if and only if ( κ i ◦ z, κ j ◦ w ) ∈ R E .It is clear that (1) implies (2) . To prove the converse direction, we observethat, as we remarked above, ( s, t ) ∈ R E if and only if for every i, j ∈ I ( κ i ◦ a si ◦ q si,j , κ j ◦ a tj ◦ q ti,j ) ∈ R E s,ti,j ; but the latter condition is equivalent, underassumption (2) , to the requirement that for every i, j ∈ I , p ◦ κ i ◦ a si ◦ q si,j = p ◦ κ j ◦ a tj ◦ q ti,j , where s ◦ r s,ti,j = κ i ◦ a si ◦ q si,j and t ◦ r s,ti,j = κ j ◦ a tj ◦ q ti,j , thatis, as the family of arrows r s,ti,j is epimorphic, to the condition p ◦ s = p ◦ t ,as required. (cid:3) Lemma 2.28.
Let B be an object of a topos E . Then an equivalence relationin E on B can be identified with an assignment E → R E of an equivalencerelation on the set Hom E ( E, B ) satisfying the following properties:(i) For any arrow f : E → E ′ in E , if ( h, k ) ∈ R E ′ then ( h ◦ f, k ◦ f ) ∈ R E ;(ii) For any epimorphic family { e i : E i → E | i ∈ I } in E and any arrows f, g ∈ Hom E ( E, B ) , ( f ◦ e i , g ◦ e i ) ∈ R E i for all i ∈ I implies ( f, g ) ∈ R E . Proof
Clearly, for any object A of C , the subobjects of A can be identifiedwith the J E -closed sieves on A , or equivalently with the c J E -closed subob-jects of the representable Hom E ( − A ) , where c J E is the closure operation onsubobjects corresponding to the canonical Grothendieck topology J E on E .Applying this remark to A = B × B and noticing that the concept of equiva-lence relation is cartesian and hence preserved and reflected by cartesian fullyfaithful functors, in particular by the Yoneda embedding E → [ E op , Set ] , weimmediately deduce the thesis. (cid:3) Lemma 2.28 implies that ( s, t ) ∈ R E if and only if for every i, j ∈ I ( κ i ◦ a si ◦ q si,j , κ j ◦ a tj ◦ q ti,j ) ∈ R E s,ti,j ; indeed, s ◦ r s,ti,j = κ i ◦ a si ◦ q si,j and t ◦ r s,ti,j = κ j ◦ a tj ◦ q ti,j .This shows that, in order to completely describe the relation R , is sufficesto consider elements of the form κ i ◦ z and κ j ◦ w , where z : E → H ( i ) and w : E → H ( j ) , and characterize when they belong to R E .38 emma 2.29. Let E be a topos, f, g : A → B arrows in E with coequalizer q : B → C and R B × B the kernel pair of q . Then for any object E of E and any elements h, k ∈ Hom E ( E, B ) , h h, k i : E → B × B factors through R if and only if there exists an epimorphic family { e i : E i → E | i ∈ I } in E such that for each i ∈ I ( h ◦ e i , k ◦ e i ) belongs to the equivalence relation onthe set Hom E ( E i , B ) generated by the set of pairs of the form ( f ◦ a, g ◦ a ) ,for a generalized element a : E i → A in E . Proof
For any object E of E , consider the equivalence relation R f,gE on theset Hom E ( E, B ) consisting of the pairs ( h, k ) with the property that thereexists an epimorphic family { e i : E i → E | i ∈ I } in E such that for each i ∈ I ( h ◦ e i , k ◦ e i ) belongs to the equivalence relation on the set Hom E ( E i , B ) generated by the set of pairs of the form ( f ◦ a, g ◦ a ) , for a generalized element a : E i → A in E . The assignment E → R f,gE clearly satisfies the conditions ofLemma 2.28 and hence defines an equivalence relation on B in E . To provethat R = R f,g , we argue as follows. Denoting by π , π : R f,g → B thecanonical projections, we have that for any E ∈ E , Hom ( E, q ) coequalizesthe arrows Hom ( E, π ) and Hom ( E, π ) ; therefore q coequalizes π and π in E . To prove that q is actually the coequalizer of π and π in E , we observethat any arrow p : B → D such that p ◦ π = p ◦ π satisfies p ◦ f = p ◦ g andhence, by definition of q , there exists a unique factorization of p through q .As q is the coequalizer of R f,g and in a topos all equivalence relations are thekernel pairs of their coequalizers, we conclude that R f,g = R , as required. (cid:3) Remark 2.30.
Under the hypotheses of the lemma, R can be characterizedas the equivalence relation on B generated by the arrows f and g , i.e. asthe smallest equivalence relation on B containing the image of the arrow h f, g i : A → B × B . Indeed, clearly R contains the image of this arrow, andif T is an equivalence relation on B containing the image of the arrow h f, g i then the coequalizer z of T factors through q and hence the kernel pair of q ,namely R , is contained in the kernel pair of z , namely T .Recall that the internal language Σ E of a topos E is the first-order signa-ture consisting of one sort p E q for any object E of E , one function symbol p f q : p A q → p B q for any arrow f : A → B in E in E and a relation symbol p R q p A q , . . . , p A n q for each subobject R A × · · · × A n in E . There isa ‘tautological’ Σ E -structure S E in E , obtained by interpreting each sort asthe corresponding object, each function symbol as the corresponding arrowand each relation symbol as the corresponding subobject.We shall use the notation x p C q to denote a variable x of sort C , andwill omit the superscript when it can be unambiguously inferred from thecontext.Lemma 2.29 can be reformulated in logical terms as follows.39 emma 2.31. Let E be a topos, f, g : A → B arrows in E with coequalizer q : B → C and R the kernel pair of q . Let ( z B , z ′ B ) ∈ G f,g be an abbreviationfor the formula z = z ′ ∨ ( ∃ x )( p f q ( x ) = z ∧ p g q ( x ) = z ′ ) ∨ ( ∃ x ′ )( p f q ( x ′ ) = z ′ ∧ p g q ( x ′ ) = z ) over Σ E . Then the geometric bi-sequent ( p R q ( y, y ′ ) ⊣⊢ y B ,y ′ B ∨ n ∈ N ( ∃ z . . . ∃ z n ( z = x ∧ z n = x ′ ∧ ∧
The interpretation of the formula ( z B , z ′ B ) ∈ G f,g is the sym-metric and reflexive closure of the relation S f,g on B given by the im-age of the arrow h f, g i : A → B × B , that is the union of the diagonalsubobject ∆ : B → B × B of B × B , S f,g B × B and the subob-ject τ ◦ S f,g of B × B given by the composite of S f,g with the exchangeisomorphism τ : B × B → B × B . The interpretation of the formula ∨ n ∈ N ( ∃ z . . . ∃ z n ( z = x ∧ z n = x ′ ∧ ∧
Let H : I → E be a functor from a small category I to a Grothendieck topos E . Then the equivalence relation R on the object ` i ∈ I H ( i ) given by the kernel pair of the canonical arrow ` i ∈ I H ( i ) → colim ( H ) satisfies the following property: for any object E of E and any pairs ( i, x ) and ( i ′ , x ′ ) in I EH , we have that ( κ i ◦ x, κ i ′ ◦ x ′ ) ∈ R E if and only if thereexists an epimorphic family { e k : E k → E | k ∈ K } in E such that for every k ∈ K , the pair (( i, x ◦ e k ) , ( i ′ , x ′ ◦ e k )) belongs to the equivalence relation onthe set I E k H generated by the pairs of the form (( j, y ) , ( j ′ , H ( f ) ◦ y )) , where f : j → j ′ is an arrow in I and y is a generalized element E k → H ( j ) . Proof
Let us consider the pair of arrows a, b : ` u : i → j in I H ( i ) → ` i ∈ I H ( i ) defined at the beginning of section 2.7. The canonical arrow ` i ∈ I H ( i ) → colim ( H ) is the coequalizer of a and b .40or any object E of E , let R E be the relation on the set Hom E ( E, ` i ∈ I H ( i )) consisting of the pairs ( x, x ′ ) with the property that for any i, j ∈ I thereexists an epimorphic family { e ( i,j ) k : E ( i,j ) k → E x,x ′ i,j | k ∈ K ( i,j ) } in E suchthat for each k ∈ K ( i,j ) , ( x ◦ r x,x ′ i,j ◦ e ( i,j ) k , x ◦ r x,x ′ i,j ◦ e ( i,j ) k ) belongs to theequivalence relation T ( i,j ) k on the set I E ( i,j ) k H generated by the set of pairs ofthe form (( n, y ) , ( m, H ( z ) ◦ y )) (for n ∈ I , y : E ( i,j ) k → H ( n ) and z : n → m in I ).It is readily seen that the assignment E → R E satisfies the hypothesesof Lemma 2.28 (notice that the fibered product of two epimorphic families isagain an epimorphic family), whence it defines an equivalence relation R in E on the object ` i ∈ I H ( i ) such that for any generalized elements x, x ′ : E → ` i ∈ I H ( i ) , h x, x ′ i factors through R if and only if ( x, x ′ ) ∈ R E .The arrow h a, b i factors through R since for every i, j ∈ I , ( a ◦ r s,ti,j , b ◦ r s,ti,j ) belongs to the equivalence relation T ( i,j ) k .By Remark 2.30, R thus contains the kernel pair of q . The converseinclusion follows from the fact that for any ( x, x ′ ) ∈ R E , q ◦ x = q ◦ x ′ ; indeed,if { e ( i,j ) k : E ( i,j ) k → E x,x ′ i,j | k ∈ K ( i,j ) } in E is an epimorphic family such thatfor each k ∈ K ( i,j ) , ( x ◦ r x,x ′ i,j ◦ e ( i,j ) k , x ◦ r x,x ′ i,j ◦ e ( i,j ) k ) belongs to the equivalencerelation T ( i,j ) k and hence we have q ◦ x ◦ r x,x ′ i,j ◦ e ( i,j ) k = q ◦ x ′ ◦ r x,x ′ i,j ◦ e ( i,j ) k forall k ∈ K ( i,j ) , that is q ◦ x = q ◦ x ′ . (cid:3) Remark 2.33.
The proposition could be alternatively deduced as a con-sequence of Lemma 2.29, but the proof of such implication would be moreinvolved than the direct one that we have given above.In the case of filtered indexing categories, the description of colimits givenabove simplifies. We shall be concerned in particular with functors of theform F ◦ π P , where F is a flat functor with values in a Grothendieck toposand π P is the fibration associated to a set-valued functor P .Recall from chapter VII of [26] that a functor F : C op → E is flat if andonly if it is filtering, i.e. the following conditions are satisfied:(i) For any object E of E there exists some epimorphic family { e i : E i → E | i ∈ I } in E and for each index i an object b i of C and a generalizedelement E i → F ( b i ) in E ;(ii) For any two objects c and d in C and any generalized element ( x, y ) : E → F ( c ) × F ( d ) in E there is an epimorphic family { e i : E i → E | i ∈ I } in E and for each index i an object b i of C with arrows u i : c → b i and v i : d → b i in C and a generalized element z i : E i → F ( b i ) in E such that h F ( u i ) , F ( v i ) i ◦ z i = h x, y i ◦ e i ;41iii) For any two parallel arrows u, v : d → c in C and any generalizedelement x : E → F ( c ) in E for which F ( u ) ◦ x = F ( v ) ◦ x , there is anepimorphic family { e i : E i → E | i ∈ I } in E and for each index i anarrow w i : c → b i and a generalized element y i : E i → F ( b i ) such that w i ◦ u = w i ◦ v and F ( w i ) ◦ y i = x ◦ e i .Notice that can suppose E = 1 E in condition ( i ) without loss of generalityand if all the arrows in the category C are monic then condition ( iii ) rewritesas follows: for any two parallel arrows u, v : D → C in C , either u = v or theequalizer of F ( u ) and F ( v ) is zero. Proposition 2.34.
Let C be a small category, E a Grothendieck topos, P : C →
Set a functor and F : C op → E a flat functor. Let π P : R P → C op bethe canonical projection from the category of elements of P to C op . Then colim ( F ◦ π P ) ∼ = ( a ( c,x ) ∈ R P F ( c )) /R, where R is the equivalence relation in E defined by saying that, for any objects ( c, x ) and ( d, y ) of the category R P , the geometric bi-sequent ( p R q ( p ξ ( c,x ) q ( u ) , p ξ ( d,y ) q ( v )) ⊣⊢ u F ( c ) ,v F ( d ) ∨ c f → a g ← d | P ( f )( x )= P ( g )( y ) ( ∃ ξ F ( a ) )( p F ( f ) q ( ξ ) = u ∧ p F ( g ) q ( ξ ) = v )) is valid in the Σ E -structure S E .In particular, for any objects ( c, x ) , ( c, y ) of the category R P , the geo-metric bi-sequent ( p R q ( p ξ ( c,x ) q ( u ) , p ξ ( c,y ) q ( u )) ⊣⊢ u F ( c ) ∨ c f → a | P ( f )( x )= P ( f )( y ) ( ∃ ξ F ( a ) )( p F ( f ) q ( ξ ) = u )) is valid in the Σ E -structure S E . Proof
Let ` ( c,x ) ∈ R P F ( c ) be the coproduct in E of the functor F ◦ π P ,with canonical coproduct arrows ξ ( c,x ) : F ( c ) → ` ( c,x ) ∈ R P F ( c ) , and let T be the equivalence relation on ` ( c,x ) ∈ R P F ( c ) corresponding to the quotient ` ( c,x ) ∈ R P F ( c ) → colim ( F ◦ π P ) . Then we have colim ( F ◦ π P ) ∼ = ( ` ( c,x ) ∈ R P F ( c )) /R , and by Lemma 2.29, the relation R is the equivalence relation on thecoproduct ` ( c,x ) ∈ R P F ( c ) generated by the image of the arrow h a, b i where s, t : ` z :( c,x ) → ( d,y ) in R P F ( c ) → ` ( c,x ) ∈ R P F ( c ) are the arrows defined above.42et λ z : F ( c ) → ` z :( c,x ) → ( d,y ) in R P F ( c ) be the canonical coproduct arrows(for each arrow z : ( c, x ) → ( d, y ) in the category R P ). Notice that for anyarrow z : ( c, x ) → ( d, y ) in R P the sequent ⊤ ⊢ u ′ F ( c ) ( p a q ( p λ z q ( u ′ )) , p b q ( p λ z q ( u ′ ))) ∈ E ( c,x ) , ( d,y ) is valid in S E .The fact that the right-to-left implication of the bi-sequent in the state-ment of the proposition holds in S E is clear.For any objects ( c, x ) , ( d, y ) in R P , let the expression ( u F ( c ) , v F ( d ) ) ∈ E ( c,x ) , ( d,y ) be an abbreviation of the formula ∨ c f → a g ← d | P ( f )( x )= P ( g )( y ) ( ∃ ξ F ( a ) )( p F ( f ) q ( ξ ) = u ∧ p F ( g ) q ( ξ ) = v )) .The validity of the left-to-right implication will clearly follow from Lemma2.31 once we have shown that for any objects ( c, x ) , ( d, y ) , ( e, z ) in R P thesequent ( u, v ) ∈ E ( c,x ) , ( d,y ) ∧ ( v, w ) ∈ E ( d,y ) , ( e,z ) ⊢ u F ( c ) ,v F ( d ) ,z F ( e ) ( u, w ) ∈ E ( c,x ) , ( e,z ) is valid in the Σ E -structure S E .For simplicity we shall give the proof in the case E = Set , but all of ourthe arguments are formalizable in the internal logic of the topos and henceare valid in general. In fact, the lift of the proof from the set-theoretic to thetopos-theoretic setting is made possible by the following logical characteriza-tion of flatness for a functor; specifically, the fact that a functor F : C op → E is flat can be expressed in terms of the validity of the following sequents inthe structure S E : ( ⊤ ⊢ [] ∨ c ∈ Ob ( C ) ( ∃ x F ( c ) )( x = x ));( ⊤ ⊢ x F ( a ) ,y F ( b ) ∨ a f → c g ← b ( ∃ z F ( c ) )( p F ( f ) q ( z ) = x ∧ p F ( g ) q ( z ) = y )) for any objects a , b of C ; ( p F ( f ) q ( x ) = p F ( g ) q ( x ) ⊢ x F ( a ) ∨ h : a → c | h ◦ f = h ◦ g ( ∃ z F ( c ) )( p F ( h ) q ( z ) = x )) for any pair of arrows f, g : b → a in C with common domain and codomain.Now, given objects ( c, x ) , ( d, y ) , ( e, z ) ∈ R P and elements u ∈ F ( c ) , v ∈ F ( d ) , w ∈ F ( e ) , suppose that ( u, v ) ∈ E ( c,x ) , ( d,y ) and ( v, w ) ∈ E ( d,y ) , ( e,z ) ; wewant to prove that ( u, w ) ∈ E ( c,x ) , ( e,z ) .As ( u, v ) ∈ E ( c,x ) , ( d,y ) , there exist arrows f : c → a and g : d → a in C and an element ξ ∈ F ( a ) such that F ( f )( ξ ) = u , F ( g )( ξ ) = v , P ( f )( x ) = P ( g )( y ) . Similarly, as ( v, w ) ∈ E ( d,y ) , ( e,z ) , there exist arrows h : d → b and k : e → b in C and an element χ ∈ F ( b ) such that F ( h )( χ ) = v , F ( k )( χ ) = w and P ( h )( y ) = P ( k )( z ) .Using condition ( ii ) in the definition of flat functor, we obtain the exis-tence of an object m of C , arrows r : a → m and t : b → m in C and an43lement ǫ ∈ F ( m ) such that F ( r )( ǫ ) = ξ and F ( t )( ǫ ) = χ . Now, considerthe arrows r ◦ g, t ◦ h : d → m ; we have that F ( r ◦ g )( ǫ ) = F ( t ◦ h )( ǫ ) andhence by condition ( iii ) in the definition of flat functor there exists an arrow s : m → n in C such that s ◦ r ◦ g = s ◦ t ◦ h and an element α ∈ F ( n ) suchthat F ( s )( α ) = ǫ .Consider the arrows s ◦ r ◦ f and s ◦ t ◦ k . They satisfy the necessaryconditions to ensure that ( u, w ) ∈ E ( c,x ) , ( e,z ) , i.e. F ( s ◦ r ◦ f )( α ) = u , F ( s ◦ t ◦ k )( α ) = w , P ( s ◦ r ◦ f )( x ) = P ( s ◦ t ◦ k )( z ) . Indeed, F ( s ◦ r ◦ f )( α ) = F ( f )( F ( r )( F ( s )( α ))) = F ( f )( F ( r )( ǫ )) = F ( f )( ξ ) = u , F ( s ◦ t ◦ k )( α ) = F ( k )( F ( t )( F ( s )( α ))) = F ( k )( F ( t )( ǫ )) = F ( k )( χ ) = w and P ( s ◦ r ◦ f )( x ) = P ( s ◦ r )( P ( f )( x )) = P ( s ◦ r )( P ( g )( y )) = P ( s ◦ r ◦ g )( y ) = P ( s ◦ t ◦ h )( y ) = P ( s ◦ t )( P ( h )( y )) = P ( s ◦ t )( P ( k )( z )) = P ( s ◦ t ◦ k )( z ) , as required.The second part of the proposition follows from the first by an easyapplication of condition ( iii ) in the definition of flat functor. (cid:3) The following proposition represents the translation of Proposition 2.34in the categorical language of generalized elements.
Proposition 2.35.
Let C be a small category, E a Grothendieck topos witha separating set S , P : C →
Set a functor and F : C op → E a flat functor.Let π P : R P → C op be the canonical projection from the category of elementsof P to C op . Then the equivalence relation R defined by the formula colim ( F ◦ π P ) ∼ = ( a ( c,x ) ∈ R P F ( c )) /R admits the following characterization: for any objects ( c, x ) , ( d, y ) of R P and generalized elements u : E → F ( c ) , v : E → F ( d ) (where E ∈ S ), ( ξ ( c,x ) ◦ u, ξ ( d,y ) ◦ v ) ∈ R E if and only if there exists an epimorphic family { e i : E i → E | i ∈ I, E i ∈ S } and for each index i ∈ I an object a i ∈ C , ageneralized element h i : E i → F ( b i ) and two arrows f i : c → a i and f ′ i : d → a i in C such that P ( f i )( x ) = P ( f ′ i )( y ) and h F ( f i ) , F ( f ′ i ) i ◦ h i = h u, v i ◦ e i .In particular, for any objects ( c, x ) and ( c, y ) of R P and any generalizedelement u : E → F ( c ) (where E ∈ S ), ( ξ ( c,x ) ◦ u, ξ ( d,y ) ◦ v ) ∈ R E if and only ifthere exists an epimorphic family { e i : E i → E | i ∈ I, E i ∈ S } and for eachindex i ∈ I an object a i ∈ C , a generalized element h i : E i → F ( a i ) and anarrow f i : c → a i in C such that P ( f i )( x ) = P ( f ′ i )( y ) and F ( f i ) ◦ h i = u ◦ e i . (cid:3) In this section, we investigate the operation on flat functors induced by ageometric morphism of toposes. This will be relevant for the characterizationof the class of theories classified by a presheaf topos addressed in section 5.44 .1 General extensions
Let ( C , J ) and ( D , K ) be Grothendieck sites, and u : Sh ( D , K ) → Sh ( C , J ) ageometric morphism. This morphism induces, via Diaconescu’s equivalence,a functor ξ E : Flat K ( D , E ) → Flat J ( C , E ) , where we write Flat Z ( R , E ) for the category of flat Z -continuous functorsfrom R to E .This functor can be described explicitly as follows. For any flat K -continuous functor F : D → E with corresponding geometric morphism f F : E → Sh ( D , K ) , the flat functor ˜ F := ξ E ( F ) : C → E is given by f ∗ F ◦ u ∗ ◦ l C , where l C : C → Sh ( C , J ) is the composite of the Yoneda embedding y C : C → [ C op , Set ] with the associated sheaf functor [ C op , Set ] → Sh ( C , J ) .For any natural transformation α : F → G between flat functors in Flat K ( D , E ) , the corresponding natural transformation f α ∗ : f F ∗ → f G ∗ ,applied to the functor u ∗ ◦ l C , gives rise to a natural transformation ˜ α : ˜ F → ˜ G which is precisely ξ E ( α ) .We can express ˜ F directly in terms of F by using a colimit construction,as follows.For any c ∈ C , the K -sheaf u ∗ ( l C ( c )) : D op → Set , considered as an objectof [ D op , Set ] , can be canonically expressed as a colimit of representables,indexed over its category of elements A c ; specifically, u ∗ ( l C ( c )) ∼ = colim ( y D ◦ π c ) in [ D op , Set ] , where π c : A c → D is the canonical projection functor. Asthe associated sheaf functor a K : [ D op , Set ] → Sh ( D , K ) preserves colimits,the functor u ∗ ( l C ( c )) is isomorphic in Sh ( D , K ) to the colimit of the functor l D ◦ π c . Therefore ˜ F ( c ) = ( f ∗ F ◦ u ∗ ◦ l C )( c ) = f ∗ F ( colim ( l D ◦ π c )) ∼ = colim ( f ∗ F ◦ l D ◦ π c ) = colim ( F ◦ π c ) .For any ( d, z ) ∈ A c , we write κ F ( d,z ) : F ( d ) → ˜ F ( c ) for the canonicalcolimit arrow. Proposition 3.1. (i) For any natural transformation α : F → G betweenflat functors F, G : D → E in Flat K ( D , E ) , the natural transformation ˜ α : ˜ F → ˜ G is characterized by the following condition: for any object ( d, z ) of the category A c the diagram ˜ F ( c ) ˜ α ( c ) / / ˜ G ( c ) F ( d ) κ F ( d,z ) O O α ( d ) / / G ( d ) κ G ( d,z ) O O commutes.(ii) The functor ξ E : Flat K ( D , E ) → Flat J ( C , E ) is natural in E ; that is, or any geometric morphism f : F → E , the diagram
Flat K ( D , E ) f ∗ ◦− (cid:15) (cid:15) ξ E / / Flat J ( C , E ) f ∗ ◦− (cid:15) (cid:15) Flat ( D , F ) ξ F / / Flat ( C , F ) commutes.(iii) For any arrow f : c → c ′ in C and any object ( d, z ) in A c , the diagram ˜ F ( c ) ˜ F ( f ) / / ˜ F ( c ′ ) F ( d ) κ F ( d,z ) O O κ F ( d,u ∗ ( l C ( f ))( z )) ; ; ①①①①①①①①① commutes. Proof ( i ) The given square commutes as it is the naturality square for thenatural transformation f α ∗ : f F ∗ → f G ∗ with respect to the canonical colimitarrow y D ( d ) → u ∗ ( l C ( c )) in [ D op , Set ] corresponding to the object ( d, z ) of A c . ( ii ) This follows as an immediate consequence of the fact that Dia-conescu’s equivalences are natural in E . ( iii ) Given a natural transformation β : P → P ′ of presheaves P and P ′ in [ D op , Set ] , denoting by κ P : R P → D and κ P ′ : R P ′ → D thecanonical projection functors and by κ P ( d,z ) : y D d → P , κ P ′ ( d ′ ,z ′ ) : y D d ′ → P thecanonical colimit arrows in [ D op , Set ] (for ( d, z ) ∈ R P and ( d ′ , z ′ ) ∈ R P ′ ),we clearly have that β ◦ κ P ( d,z ) = k P ′ ( d,β ( d )( z )) . To obtain our thesis it sufficesto apply this result to the natural transformation β : u ∗ ( l C ( f )) : u ∗ ( l C ( c )) → u ∗ ( l C ( c ′ )) ; indeed, by applying the functor f ∗ F to the resulting equality weobtain precisely the identity in the statement of the proposition. (cid:3) Let D be a subcategory of a small category C . Then the inclusion functor i : D ֒ → C induces a geometric morphism E ( i ) : [ D , Set ] → [ C , Set ] andhence, by Diaconescu’s equivalence, a functor ξ E : Flat ( D op , E ) → Flat ( C op , E ) . We can describe ˜ F directly in terms of F as follows. For any c ∈ C , thefunctor E ( i ) ∗ ( y C ( c )) : D →
Set coincides with the functor
Hom C ( c, − ) : D →
Set . Its category of elements A c has as objects the pairs ( d, h ) where46 is an object of D and h is an arrow h : c → d in C and as arrows ( d, h ) → ( d ′ , h ′ ) the arrows k : d ′ → d in D such that k ◦ h ′ = h in C . As f ∗ F preserves allsmall colimits and Hom C ( c, − ) is the colimit in [ D , Set ] of the compositionof the canonical projection π c : A c → D op with the Yoneda embedding D op → [ D , Set ] , the object ˜ F ( c ) can be identified with the colimit of thecomposite functor F ◦ π c : A c → E . For any object d of D , the pair ( d, d ) defines an object of the category A d and hence a canonical colimit arrow χ d : F ( d ) → ˜ F ( d ) , also denoted by χ Fd . For any object ( d, h ) of A c , thecolimit arrow κ ( d,h ) : F ( d ) → ˜ F ( c ) is equal to the composite ˜ F ( h ) ◦ χ d . Remarks 3.2. (i) The functor ξ E : Flat ( D op , E ) → Flat ( C op , E ) coin-cides with the (restriction to the subcategories of flat functors) ofthe left Kan extension functor [ D op , E ] → [ C op , E ] along the inclusion D op ֒ → C op (cf. Remark 3.13 below).(ii) For any object d ∈ D the diagram ˜ F ( d ) ˜ α ( d ) / / ˜ G ( d ) F ( d ) χ Fd O O α ( d ) / / G ( d ) χ Gd O O commutes (cf. Proposition 3.1(a)).(iii) For any object d ∈ D , the arrow χ d : F ( d ) → ˜ F ( d ) is monic. Indeed, itcan be identified with the image of the canonical subfunctor inclusion Hom D ( d, − ) ֒ → Hom C ( d, − ) under the inverse image f ∗ F of the geo-metric morphism f F . From this, in view of Remark 3.2(a), it followsat once that the functor ξ E : Flat ( D op , E ) → Flat ( C op , E ) is faithful.(iv) For any flat functors F, G : D op → E and natural transformation β :˜ F → ˜ G , β = ˜ α for some natural transformation α : F → G if and onlyif for every d ∈ D the diagram ˜ F ( d ) β ( d ) / / ˜ G ( d ) F ( d ) χ Fd O O α ( d ) / / G ( d ) χ Fd O O commutes. One direction follows from Remark 3.2(a). To prove theother one we observe that β ( c ) = ˜ α ( c ) for all c ∈ C if and only if forevery object ( a, z ) ∈ A c , β ( c ) ◦ κ ( a,z ) = ˜ α ( c ) ◦ κ ( a,z ) . Now, β ( c ) ◦ κ ( a,z ) = β ( c ) ◦ ˜ F ( z ) ◦ χ Fa = ˜ G ( z ) ◦ β ( a ) ◦ χ Fa = ˜ G ( z ) ◦ χ Ga ◦ α ( a ) = κ G ( a,z ) ◦ α ( a ) =˜ α ( c ) ◦ κ F ( a,z ) (where the third equality follows from the naturality of β and the last from Proposition 3.1(a)), as required.47v) Let d be an object of D and y D d = Hom D ( − , d ) : D op → Set therepresentable on D associated to d . Then ξ E ( y D d ) = y C d , where y C d = Hom C ( − , d ) is the representable on C associated to d (considered hereas an object of C ). Indeed, the flat functor y D d corresponds to thegeometric morphism whose inverse image is the evaluation functor ev D d :[ D , Set ] → Set at the object d , and the composite ev D d ◦ E ( i ) ∗ coincideswith the evaluation functor ev C d : [ C , Set ] → Set at the object d , whichcorresponds to the flat functor y C d .(vi) Let E be a Grothendieck topos and γ E : E →
Set the unique ge-ometric morphism from E to Set . Then by Proposition 3.1(ii) andRemark 3.2(v), for any object d of D , the functor ξ E : Flat ( D op , E ) → Flat ( C op , E ) sends the flat functor γ ∗E ◦ y D d to the functor γ ∗E ◦ y C d . Proposition 3.3.
Under the natural equivalences e C : Ind - C ≃
Flat ( C op , Set ) and e D : Ind - D ≃
Flat ( D op , Set ) , the functor ξ Set corresponds to the functor
Ind - i : Ind - D →
Ind - C . Proof
Let us first recall the definition of the functor
Ind - i : Ind - D →
Ind - C .For any flat functor F : C op → Set , Ind - i ( F ) is the functor given by thecolimit in Ind - C of the composite functor i ◦ π F , where π F : R F → D is thecanonical projection from the category of elements R F of F to D . For any c ∈ C , we thus have Ind - i ( F )( c ) = colim ( ev c ◦ i ◦ π F ) , where ev c : Ind - C →
Set is the evaluation functor at c .Now, the functor ev c ◦ i : D →
Set is equal to the functor
Hom C ( c, − ) : D →
Set considered above, and ˜ F ( c ) = colim ( F ◦ π c ) : A c → Set , where A c is the category of elements of ev c ◦ i and π c : A c → D op the associatedcanonical projection. The commutativity of the tensor product betweena presheaf and a covariant set-valued functor (cf. chapter VII of [26] orTheorem 2.5 above) thus yields a natural isomorphism between the two sets,as required. (cid:3) Corollary 3.4.
Let D ֒ → C be an embedding of small categories and F : D op → E a flat functor. With the above notation, for any object c ∈ C , wehave ˜ F ( c ) ∼ = a ( a,z ) ∈A c F ( c ) /R c , where R c is the equivalence relation in E defined by saying that, for anyobjects ( a, z ) , ( a ′ , z ′ ) of the category A c , the geometric bi-sequent ( p R q ( p ξ ( a,z ) q ( x ) , p ξ ( a ′ ,z ′ ) q ( x ′ )) ⊣⊢ x F ( a ) ,x ′ F ( a ′ ) ∨ a f → b g ← a ′ | f ◦ z = g ◦ z ′ ( ∃ y F ( b ) )( p F ( f ) q ( y ) = x ∧ p F ( g ) q ( y ) = x ′ )) s valid in the tautological Σ E -structure S E (defined in section 2.7), wherefor any object ( a, z ) of the category A c , ξ ( a,z ) : F ( a ) → ˜ F ( c ) is the canonicalcolimit arrow.In particular, for any objects ( a, z ) , ( a, z ′ ) of the category A c , the geomet-ric bi-sequent ( p R q ( p ξ ( a,z ) q ( x ) , p ξ ( a,z ′ ) q ( x )) ⊣⊢ x F ( a ) ∨ a f → b | f ◦ z = f ◦ z ′ ( ∃ y F ( b ) )( p F ( f ) q ( y ) = x )) is valid in the Σ E -structure S E .Semantically, the relation R c can be characterized by saying that, forany objects ( a, z ) , ( a ′ , z ′ ) of the category A c and any generalized elements x : E → F ( a ) , x ′ : E → F ( a ′ ) , ( κ ( a,z ) ◦ x, κ ( a ′ ,z ′ ) ◦ x ′ ) ∈ R c if and onlyif there exists an epimorphic family { e i : E i → E | i ∈ I } and for eachindex i ∈ I an object b i ∈ D , a generalized element h i : E i → F ( b i ) andtwo arrows f i : a → b i and f ′ i : a ′ → b i in D such that f ′ i ◦ z ′ = f i ◦ z and h F ( f i ) , F ( f ′ i ) i ◦ h i = h x, x ′ i ◦ e i .In particular, for any objects ( a, z ) and ( a, z ′ ) of A c and any generalizedelement x : E → F ( a ) , ( κ ( a,z ) ◦ x, κ ( a,z ′ ) ◦ x ) ∈ R c if and only if there exists anepimorphic family { e i : E i → E | i ∈ I } and for each index i ∈ I an object b i ∈ D , a generalized element h i : E i → F ( b i ) and an arrow f i : a → b i in D such that f i ◦ z ′ = f i ◦ z and F ( f i ) ◦ h i = x ◦ e i . Proof
The corollary can be obtained by applying Proposition 2.34 (and itscategorical reformulation provided by Proposition 2.35) to the flat functor F : D op → E and the functor P : D →
Set given by ev c ◦ i = Hom C ( c, − ) : D →
Set , whose category of elements coincides with A c and whose associatedfibration π P coincides with π c : A c → D op . (cid:3) Let T be a geometric theory and K a small category of set-based T -models.Then the family of geometric morphisms Set → Sh ( C T , J T ) correspondingto the T -models in K induces a geometric morphism p K : [ K , Set ] → Sh ( C T , J T ) whose associated T -model in [ K , Set ] is given, at each sort, by the corre-sponding forgetful functor.We thus have, for each Grothendieck topos E , an induced functor u T ( K , E ) : Flat ( K op , E ) → Flat J T ( C T , E ) , which the following theorem describes explicitly.49efore stating it, we need to introduce some notation. For any any object { ~x . φ } of C T , we write A K{ ~x.φ } or simply A { ~x.φ } when the category K canbe obviously inferred from the context, for the category whose objects arethe pairs ( M, w ) , where M ∈ K and w ∈ [[ ~x . φ ]] M and whose arrows ( M, w ) → ( N, z ) are the T -model homomorphisms g : N → M in K suchthat g ( w ) = z . We denote by π K{ ~x.φ } : A { ~x.φ } → K op the canonical projectionfunctor. Theorem 3.5.
Let T be a geometric theory. Then for any flat functor F : K op → E , the functor ˜ F := u T ( K , E ) ( F ) : C T → E sends any object { ~x . φ } of C T to the colimit colim ( F ◦ π K{ ~x.φ } ) and acts on the arrows in the obviousway. In particular, for any formula { ~x . φ } presenting a T -model M { ~x.φ } in K , u T ( K , E ) ( F )( { ~x . φ } ) ∼ = F ( M { ~x.φ } ) . Proof
Let g F : E → [ K , Set ] be the geometric morphism corresponding, viaDiaconenscu’s equivalence, to the flat functor F . Then the functor u T ( K , E ) ( F ) is equal to the composite g ∗ F ◦ p ∗K ◦ y , where y : C T → Sh ( C T , J T ) is the Yonedaembedding.Now, for any geometric formula { ~x . φ } over Σ , p ∗K ( y ( { ~x . φ } )) is thefunctor F { ~x.φ } sending to any model M ∈ K the set [[ ~x . φ ]] M . This functorcan be expressed as the colimit colim ( y ′ ◦ π { ~x.φ } ) , where y ′ : K op → [ K , Set ] isthe Yoneda embedding, since the functor π { ~x.φ } coincides with the canonicalprojection from the category of elements of the functor F { ~x.φ } to K op .Therefore u T ( K , E ) ( F )( { ~x . φ } ) = g ∗ F ( colim ( y ′ ◦ π { ~x.φ } )) ∼ = colim ( g ∗ F ◦ y ′ ◦ π { ~x.φ } ) ∼ = colim ( F ◦ π { ~x.φ } ) , as required. (cid:3) Remark 3.6.
From the proof of the theorem, it is clear that the isomor-phism u T ( K , E ) ( F )( { ~x . φ } ) ∼ = F ( M { ~x.φ } ) is natural in { ~x . φ } ; that is, for anygeometric formulae { ~x . φ } and { ~y . ψ } respectively presenting T -models M { ~x.φ } and M { ~y.ψ } and any T -provably functional formula θ : { ~x . φ } →{ ~y . ψ } inducing a T -model homomorphism M θ : M { ~y.ψ } → M { ~x.φ } , thearrow u T ( K , E ) ( F )( θ ) is given by the image of the arrow F ( M θ ) across theisomorphisms u T ( K , E ) ( F )( { ~x . φ } ) ∼ = F ( M { ~x.φ } ) and u T ( K , E ) ( F )( { ~y . ψ } ) ∼ = F ( M { ~y.ψ } ) . Corollary 3.7.
Let T be a geometric theory, σ = ( φ ⊢ ~x ψ ) a geometricsequent over the signature of T and F : K op → E a flat functor. If σ isvalid in every T -model in K then ˜ F ( { ~x . φ } ) ≤ ˜ F ( { ~x . ψ } ) as subobjects of ˜ F ( { ~x . ⊤} ) in E . Proof
By Theorem 3.5, we have that ˜ F ( { ~x . φ } ) = colim ( F ◦ π { ~x.φ } ) , ˜ F ( { ~x . ψ } ) = colim ( F ◦ π { ~x.ψ } ) and ˜ F ( { ~x . ⊤} ) = colim ( F ◦ π { ~x. ⊤} ) . Now, A { ~x.φ } and A { ~x.ψ } canonically embed as subcategories of A { ~x. ⊤} , and if σ is50alid in every T -model in K then we have a canonical functor i : A { ~x.φ } →A { ~x.ψ } which commutes with these embeddings. It thus follows from thefunctoriality of colimits that ˜ F ( { ~x . φ } ) ≤ ˜ F ( { ~x . ψ } ) as subobjects of ˜ F ( { ~x . ⊤} ) in E , as required. (cid:3) Let us now apply Proposition 2.35 in the context of extensions F → ˜ F of flat functors induced by the geometric morphism p K : [ K , Set ] → Sh ( C T , J T ) . The following characterization is obtained by applying it in conjunctionwith Proposition 4.5.
Proposition 3.8.
Let T be a geometric theory over a signature Σ , K a smallcategory of set-based T -models, E a Grothendieck topos with a separating set S and F : K op → E a flat functor. With the above notation, for any geometricformula-in-context φ ( ~x ) over Σ , we have ˜ F ( { ~x . φ } ) ∼ = ( a ( M,z ) ∈A { ~x.φ } F ( M )) /R { ~x.φ } , where R { ~x.φ } is the equivalence relation in E defined by saying that for anyobjects ( M, z ) , ( N, w ) of the category A { ~x.φ } and any generalized elements x : E → F ( M ) , x ′ : E → F ( N ) (where E ∈ S ), we have ( ξ ( M,z ) ◦ x, ξ ( N,w ) ◦ x ′ ) ∈ R { ~x.φ } if and only if there exists an epimorphic family { e i : E i → E | i ∈ I, E i ∈ S } and for each index i ∈ I a T -model a i in K , a generalizedelement h i : E i → F ( a i ) and two T -model homomorphisms f i : M → a i and f ′ i : N → a i such that f i ( z ) = f ′ i ( w ) and h F ( f i ) , F ( f ′ i ) i ◦ h i = h x, x ′ i ◦ e i (where for any object ( M, z ) of the category A { ~x.φ } , κ ( M,z ) : F ( M ) → ˜ F ( { ~x .φ } ) is the canonical colimit arrow).In particular, for any objects ( M, z ) and ( M, w ) of A { ~x.φ } and any gener-alized element x : E → F ( M ) (where E ∈ S ), ( κ ( M,z ) ◦ x, κ ( M,w ) ◦ x ) ∈ R { ~x.φ } if and only if there exists an epimorphic family { e i : E i → E | i ∈ I, E i ∈ S } and for each index i ∈ I a T -model a i in K , a generalized element h i : E i → F ( a i ) and a T -model homomorphism f i : M → a i such that f i ( z ) = f i ( w ) and F ( f i ) ◦ h i = x ◦ e i . (cid:3) Let M T the universal model of T in the syntactic category C T . We canrepresent the model ˜ F ( M T ) as a E -indexed filtered colimit of set-based mod-els of T . For simplicity, we shall first establish this representation in the caseof the topos of sets, and then generalize it to an arbitrary topos.Let F : K op → Set be a flat functor. By Theorem 3.5, for any sort A over Σ , ˜ F ( M T ) A = ˜ F ( { x A . ⊤} ) = colim ( F ◦ π { x A . ⊤} ) , where π { x A . ⊤} isthe canonical projection functor A { x A . ⊤} → K op to K op from the category51f elements A { x A . ⊤} of the functor P A : K →
Set which assigns any model M in K to the set M A and acts accordingly on the arrows. Now, it followsfrom the commutativity of the tensor product between a presheaf and acovariant set-valued functor (cf. chapter VII of [26] or section 2.5 above) thatthe colimit colim ( F ◦ π { x A . ⊤} ) is isomorphic to the colimit colim ( P A ◦ π F ) ,where π F : R F → K is the canonical projection functor from the categoryof elements of the functor F to K . Since the functor F is flat, the category R F is filtered. Therefore, as filtered colimits in T -mod ( Set ) are computedsortwise as in Set , ˜ F ( M T ) ∼ = colim ( i ◦ π F ) , where i : K ֒ → T -mod ( Set ) is thecanonical inclusion. So for any object ( c, x ) of the category R F we have a T -model homomorphism ξ ( c,x ) : c → ˜ F ( M T ) , which can be expressed in termsof the colimit arrows κ ( a,y ) : F ( a ) → ˜ F ( { x A . ⊤} ) = ˜ F ( M T ) A (for y ∈ cA and A sort over Σ ) as follows: for any sort A over Σ , ξ ( c,x ) A ( y ) = κ ( c,y ) ( x ) .The explicit description of filtered colimits in the category Set thus yields,for each sort A over Σ , the following characterizing properties of the colimit colim ( P A ◦ π F ) in terms of the arrows ξ ( c,x ) :(i) For any element x of ˜ F ( M T ) A there exists an object ( c, x ) of the cate-gory R F and an element y of cA such that ξ ( c,x ) A ( y ) = x ;(ii) For any objects ( c, x ) and ( c ′ , x ′ ) of the category R F and elements y ∈ cA and y ′ ∈ c ′ A , we have ξ ( c,x ) A ( y ) = ξ ( c ′ ,x ′ ) A ( y ′ ) if and only if thereexists an object ( c ′′ , x ′′ ) of R F and arrows f : c → c ′′ and g : c ′ → c ′′ in K such that F ( f )( x ′′ ) = x , F ( g )( x ′′ ) = x ′ and f A ( y ) = gA ( y ′ ) .Moreover, the filteredness of the category R F implies the following ‘jointembedding property’: for any objects ( c, x ) and ( c ′ , x ′ ) of the category R F there exists an object ( c ′′ , x ′′ ) of R F and arrows f : c → c ′′ and g : c ′ → c ′′ in K such that F ( f )( x ′′ ) = x , F ( g )( x ′′ ) = x ′ and ξ ( c,x ) ◦ f = ξ ( c,x ) ◦ g .Let us now proceed to establish the E -indexed generalization of this re-sult.Suppose that F : K op → E is a flat functor, where K is a small subcat-egory of the category T -mod ( Set ) . By Theorem 3.5, for any sort A over Σ , we have that ˜ F ( M T ) A = ˜ F ( { x A . ⊤} ) = colim ( F ◦ π { x A . ⊤} ) ; in partic-ular, for any object ( a, y ) of the category A { x A . ⊤} we have a colimit arrow κ ( a,y ) : F ( a ) → ˜ F ( { x A . ⊤} ) = ˜ F ( M T ) A in E . More generally, for anyformula-in-context φ ( ~x ) over Σ , any model c in K and any element ~y of [[ ~x . φ ]] c , we have a colimit arrow κ ( c,~y ) : F ( c ) → ˜ F ( { ~x . φ } ) . Proposition 3.9.
Let T be a geometric theory over a signature Σ , K asmall subcategory of the category T -mod ( Set ) , E a Grothendieck topos and F : K op → E a flat functor. With the above notation, for any pair ( c, x ) con-sisting of an object c of K and of a generalized element x : E → F ( c ) , thereis a Σ -structure homomorphism ξ ( c,x ) : c → Hom E ( E, ˜ F ( M T )) defined as ollows: for any sort A over Σ , the function ξ ( c,x ) : cA → Hom E ( E, ˜ F ( { x A . ⊤} )) sends any element y ∈ cA to the generalized element κ ( c,y ) ◦ x : E → ˜ F ( { x A . ⊤} ) . Proof
We have to verify that:(1) For any function symbol f : A , . . . , A n → B over Σ , the diagram cA × · · · × cA nξ ( c,x ) A ×···× ξ ( c,x ) A n (cid:15) (cid:15) cf / / cB ξ ( c,x ) B (cid:15) (cid:15) Hom E ( E, ˜ F ( { x A . ⊤} ) × · · · × ˜ F ( { x A n . ⊤} )) i (cid:15) (cid:15) Hom E ( E, ˜ F ( { x A , . . . , x A n . ⊤} )) Hom E ( E, ˜ F ([ f ])) / / Hom E ( E, ˜ F ( { x B . ⊤} )) , where i is the canonical isomorphism ˜ F ( { x A . ⊤} ) × · · · × ˜ F ( { x A n . ⊤} ) ∼ = ˜ F ( { x A , . . . , x A n . ⊤} ) induced by the preservation of finite prod-ucts by ˜ F , commutes;(2) For any relation symbol R A , . . . , A n over Σ , we have a commutativediagram cR (cid:15) (cid:15) / / Hom E ( E, ˜ F ( { x A , . . . , x A n . R )) (cid:15) (cid:15) cA × · · · × cA n / / Hom E ( E, ˜ F ( { x A , . . . , x A n . ⊤} )) , where the unnamed vertical arrows are the canonical ones and the lowerhorizontal arrow is Hom E ( E, i ) ◦ ξ ( c,x ) A × · · · × ξ ( c,x ) A n .To prove (1) , we first observe that for any n -tuple ~y = ( y , . . . , y n ) ∈ cA ×· · ·× cA n , i ◦h κ ( c,y ) , . . . , κ ( c,y n ) i = κ ( c,~y ) . Indeed, for any i ∈ { , . . . , n } ,the arrow ˜ F ( π i ) ◦ i , where π i : { x A , . . . , x A n . ⊤} → { x A i . ⊤} is thecanonical projection arrow in the syntactic category C T , is equal to the i -th product projection ˜ F ( { x A . ⊤} ) × · · · × ˜ F ( { x A n . ⊤} ) → ˜ F ( { x A i . ⊤} ) , by Proposition 3.1(iii). Therefore, to prove the required condition it isequivalent to verify that for any n -tuple ~y = ( y , . . . , y n ) ∈ cA × · · · × cA n , ξ ( c,x ) B ( cf ( ~y )) = ˜ F ([ f ]) ◦ κ ( c,~y ) ◦ x , where [ f ] : { x A , . . . , x A n . ⊤} → { x B . ⊤} is the arrow in the syntactic category C T corresponding to the functionsymbol f . But ξ ( c,x ) B ( cf ( ~y )) = κ ( c,cf ( ~y )) ◦ x , and we have that ˜ F ([ f ]) ◦ κ ( c,~y ) = κ ( c,cf ( ~y )) , again by Proposition 3.1(iii).To prove (2) , it suffices to observe that, by Proposition 3.1(iii), for any n -tuple ~y = ( y , . . . , y n ) ∈ cA × · · · × cA n in cR , κ ( c,~y ) factors through thecanonical subobject ˜ F ( R ) ˜ F ( { x A , . . . , x A n . ⊤} ) . (cid:3) ξ ( c,x ) . Lemma 3.10.
Let T be a geometric theory over a signature Σ , K a smallsubcategory of the category T -mod ( Set ) and F : K op → E a flat functor withvalues in a Grothendieck topos E . Then(i) For any generalized element x : E → F ( c ) and any arrow f : d → c in K , ξ ( c,x ) ◦ f = ξ ( c,F ( f ) ◦ x ) ;(ii) For any generalized element x : E → F ( c ) and any arrow e : E ′ → E , ξ ( c,x ◦ e ) = Hom ( e, ˜ F ( M T )) ◦ ξ ( c,x ) Proof
These properties can be easily proved by using the definition of thearrows ξ ( c,x ) in terms of the arrows κ ( a,y ) and Proposition 3.1(iii). (cid:3) Proposition 3.11.
Let T be a geometric theory over a signature Σ , K a smallsubcategory of the category T -mod ( Set ) and F : K op → E a flat functor withvalues in a Grothendieck topos E . Let M be the T -model ˜ F ( M T ) in E . Then,for any sort A over Σ , we have that(i) For any generalized element x : E → M A there exists an epimorphicfamily { e i : E i → E | i ∈ I } in E , for each index i ∈ I a T -model a i in K , a generalized element x i : E i → F ( a i ) and an element y i ∈ a i A such that ξ ( a i ,x i ) A ( y i ) = x ◦ e i .(ii) For any pairs ( a, x ) and ( b, x ′ ) , where a and b are T -models in K and x : E → F ( a ) , x ′ : E → F ( b ) are generalized elements, and anyelements y ∈ aA and y ′ ∈ bA , we have that ξ ( a,x ) A ( y ) = ξ ( b,x ′ ) A ( y ′ ) ifand only if there exists an epimorphic family { e i : E i → E | i ∈ I } in E in E and for each index i ∈ I a T -model c i in K , arrows f i : a → c i , g i : b → c i and a generalized element x i : E i → F ( c i ) such that h x, x ′ i ◦ e i = h F ( f i ) , F ( g i ) i ◦ x i for all i ∈ I and f i A ( y ) = g i A ( y ′ ) .Moreover, the following ‘joint embedding property’ holds: for any pairs ( a, x ) and ( b, x ′ ) , where a and b are T -models in K and x : E → F ( a ) , x ′ : E → F ( b ) are generalized elements, there exists an epimorphic family { e i : E i → E | i ∈ I } in E in E and for each index i ∈ I a T -model c i in K ,arrows f i : a → c i , g i : b → c i in K and a generalized element x i : E i → F ( c i ) such that h x, x ′ i ◦ e i = h F ( f i ) , F ( g i ) i ◦ x i for all i ∈ I and (by Lemma 3.10)the following diagram commutes: f (cid:29) (cid:29) ❁❁❁❁❁❁❁❁❁ ξ ( a,x ) / / Hom E ( E, M ) Hom E ( e i ,M ) ( ( ◗◗◗◗◗◗◗◗◗◗◗◗◗ c i ξ ( ci,xi ) / / Hom E ( E i , M ) b g A A ✂✂✂✂✂✂✂✂ ξ ( b,x ′ ) / / Hom E ( E, M ) Hom E ( e i ,M ) ♠♠♠♠♠♠♠♠♠♠♠♠♠ In fact, such family can be taken to be the pullback of the family of arrows h F ( f ) , F ( g ) i : F ( c ) → F ( a ) × F ( b ) (for all spans ( f : a → c, g : b → c ) inthe category K ) along the arrow h x, x ′ i : E → F ( a ) × F ( b ) . Proof
Recall from Theorem 3.5 that for any sort A over Σ , ˜ F ( M T ) A = colim ( F ◦ π K{ x A . ⊤} ) ; now, π K{ x A . ⊤} is precisely the functor P A : K →
Set sending a model N in C to the set N A .The proposition then straightforwardly follows from Theorem 2.26, ap-plied to the functors P A : K →
Set (with A varying among the sorts of Σ ),in view of Theorems 2.5 and 2.14. (cid:3) Let C be a small category and E a Grothendieck topos. Recall from [5] thatthe indexed category [ C , E ] E is locally small, that is for any two functors F, G : C → E there exists an object
Hom E [ C , E ] E ( F, G ) of E satisfying theuniversal property that for any object E of E the generalized elements E → Hom E [ C , E ] E ( F, G ) correspond bijectively, naturally in E ∈ C , to the naturaltransformations ! ∗ E ◦ F → ! ∗ E ◦ G .The following theorem establishes a general adjunction between cate-gories of E -valued functors induced by a functor P : C → [ D op , Set ] .Before stating it, we need to introduce some notation. For any object c of C , we denote by R P ( c ) the category of elements of the functor P ( c ) : D op → Set and by π c : R P ( c ) → D the canonical projection functor. Wedenote by y D : D op → [ D , Set ] the Yoneda embedding. For any functor H : D → E and any object ( d, z ) of the category R P ( c ) , we denote by κ ( d,z ) : H ( d ) → colim ( H ◦ π c ) the canonical colimit arrow. Theorem 3.12.
Let C and D small categories, E a Grothendieck topos and P : C → [ D op , Set ] a functor. Let ˜( − ) : [ D , E ] → [ C , E ] be the functor sendingto any functor F : D → E the functor ˜ F defined by:for any c ∈ C , ˜ F ( c ) = colim ( F ◦ π c ) , andfor any arrow f : c → c ′ , ˜ F ( f ) : colim ( F ◦ π c ) → colim ( F ◦ π c ′ ) isdefined by the conditions ˜ F ( f ) ◦ κ F ( d,z ) = κ F ( d,P ( f )( z )) (for any object ( d, z ) ofthe category R P ( c ) ), nd which acts on arrows in the obvious way.Let ( − ) r : [ C , E ] → [ D , E ] be the functor assigning to any functor G : C → E the functor
Hom E [ C , E ] E ( γ ∗E ◦ ˜ y D − , G ) , and acting on the arrows in theobvious way.Then the functors ˜( − ) : [ D , E ] → [ C , E ] and ( − ) r : [ C , E ] → [ D , E ] are adjoint to each other ( ˜( − ) on the left and ( − ) r on the right). The unit η : F → ( ˜ F ) r and the counit ǫ G : ˜( G r ) → G are defined as follows:For any d ∈ D , η F ( d ) : F ( d ) → ( ˜ F ) r ( d ) is the arrow in E defined bymeans of generalized elements by saying that it sends any generalized element E → F ( d ) , regarded as a natural transformation γ ∗E /E ◦ y D d → ! ∗ E ◦ F , to theimage of this arrow γ ∗E /E ◦ ˜ y D d → ! ∗ E ◦ ˜ F under the functor ˜( − ) , regarded asa generalized element E → Hom E [ C , E ] E ( γ ∗E ◦ ˜ y ′D d, ˜ F ) = ( ˜ F ) r ( d ) .For any c ∈ C , ǫ G ( c ) : ˜( G r )( c ) = colim ( G r ◦ π c ) → G ( c ) is definedby setting, for each object ( d, z ) of R P ( c ) , ǫ G ( c ) ◦ κ G r ( d,z ) equal to the arrow G r ( d ) → G ( c ) defined by means of generalized elements by saying that a gen-eralized element x : E → G r ( d ) , corresponding to a natural transformation x :! ∗ E ◦ γ ∗E ◦ ˜ y D d ∼ = γ ∗E /E ◦ ˜ y D d → ! ∗ E ◦ ˜ G , is sent to the arrow E → G ( c ) obtainedby composing x ( c ) with the component of γ ∗E /E ( ˜ y D d ( c )) corresponding to theelement of ˜ y D d ( c ) given by the image of the identity on d via the colimitarrow κ y D d ( d,z ) : ( y D d )( d ) → ˜ y D d ( c ) . Proof
For simplicity we shall prove the result only in the case E = Set ,the proof of the general case being entirely analogous (the only care that onehas to take is to use generalized elements in place of standard set-theoreticelements).We shall define a bijective correspondence between the natural trans-formations ˜ F → G and the natural transformations F → G r , natural in F ∈ [ D , Set ] and G ∈ [ C , Set ] .Given a natural transformation α : F → G r , we define τ ( α ) : ˜ F → G by setting, for each c ∈ C , τ α ( c ) : ˜ F ( c ) → G ( c ) equal to the arrow de-fined as follows. As ˜ F ( c ) is the colimit of the cone { κ F ( d,z ) : F ( d ) → ˜ F ( c ) | ( d, z ) ∈ R P ( c ) } , it suffices to define an arrow u ( d,z ) : F ( d ) → G ( c ) for each pair ( d, z ) where d ∈ D and z ∈ P ( c )( d ) , checking that when-ever ( d, z ) and ( d ′ , z ′ ) are pairs such that for an arrow f : d → d ′ in D , P ( c )( f )( z ′ ) = z then u ( d ′ ,z ′ ) ◦ F ( f ) = u ( d,z ) ; indeed, by the universalproperty of the colimit, such a family of arrows will induce a unique ar-row τ α ( c ) : ˜ F ( c ) → G ( c ) such that τ α ( c ) ◦ κ F ( d,z ) = u ( d,z ) (for each ( d, z ) R P ( c ) ). We set u ( d,z ) : F ( d ) → G ( c ) equal to the function sending ev-ery element x ∈ F ( d ) to the element α ( d )( x )( c )( κ y ′D d ( d,z ) (1 d )) . Let us checkthat the compatibility condition is satisfied. Given an arrow f : d → d ′ such that P ( c )( f )( z ′ ) = z , we have to verify that for every x ∈ F ( d ) , α ( d ′ )( F f ( x ))( c )( κ y ′D d ′ ( d ′ ,z ′ ) (1 d ′ )) = α ( d )( x )( c )( κ y ′D d ( d,z ) (1 d )) . This identity imme-diately follows from the commutativity of the naturality diagram F ( d ) F ( f ) (cid:15) (cid:15) α ( d ) / / G r ( d ) = Hom [ C , Set ] ( ˜ y D d, G ) G r ( f )= −◦ ˜ y D f (cid:15) (cid:15) F ( d ′ ) α ( d ′ ) / / G r ( d ′ ) = Hom [ C , Set ] ( ˜ y D d ′ , G ) for α with respect to the arrow f and that of the diagram ˜ y ′D d ′ ( c ) f / / ˜ y ′D d ( c ) y D d ′ κ y D d ′ ( d ′ ,z ′ ) O O ( y D f )( d ′ ) / / y D d, κ y D d ( d ′ ,z ′ ) O O which is an instance of Proposition 3.1(i).Indeed, α ( d ′ )( F f ( x ))( c )( κ y D d ′ ( d ′ ,z ′ ) (1 d ′ )) = α ( d )( x )( ˜ y D f ( c )( κ y D d ′ ( d ′ ,z ′ ) (1 d ′ ))) = α ( d )( x )( κ y ′D d ( d ′ ,z ′ ) ( f )) = α ( d )( x )( c )( κ y ′D d ( d,z ) (1 d )) , where the first equality followsfrom the commutativity of the first diagram, the second follows from thecommutativity of the second diagram, and the last follows from the fact thatthe family { κ y D d ( d,z ) | ( d, z ) ∈ R P ( c ) } is a cocone.To complete the definition of τ ( α ) , it remains to check that the assign-ment c → τ ( α )( c ) defines a natural transformation ˜ F → G . We have toverify that for any arrow f : c → c ′ in C , the diagram ˜ F ( c ) ˜ F ( f ) (cid:15) (cid:15) τ ( α )( c ) / / G ( c ) G ( f ) (cid:15) (cid:15) ˜ F ( c ′ ) τ ( α )( c ′ ) / / G ( c ′ ) commutes.As, by definition of ˜ F ( f ) , the diagram ˜ F ( c ) ˜ F ( f ) / / ˜ F ( c ′ ) F ( d ) κ F ( d,z ) O O κ F ( d,P ( f )( z )) ; ; ①①①①①①①①① κ F ( d,z ) are jointly epimorphic (as they are colimitarrows), the square above commutes if and only if the arrows G ( f ) ◦ τ ( α )( c ) ◦ κ F ( d,z ) and τ ( α )( c ′ ) ◦ κ F ( d,P ( f )( z )) are equal, that is if and only if they take thesame values at any element x ∈ F ( d ) . Let us set z ′ = P ( f )( d )( z ) . By defini-tion of τ ( α ) , τ ( α )( c )( κ F ( d,z ) ( x )) = α ( d )( x )( c )( κ y D d ( d,z ) (1 d )) , while τ ( α )( c ′ )( κ F ( d,z ′ ) ( x )) = α ( d )( x )( c ′ )( κ y D d ( d,z ′ ) (1 d )) . Now, for any d ∈ D and any x ∈ F ( d ) , α ( d )( x ) is anatural transformation ˜ yd → G ; in particular, the diagram ˜ yd ( c ) ˜ yd ( f ) (cid:15) (cid:15) α ( d )( x )( c ) / / G ( c ) G ( f ) (cid:15) (cid:15) ˜ yd ( c ′ ) κ F ( d,P ( f )( z )) / / G ( c ′ ) commutes.The commutativity of this diagram, together with that of the diagram ˜ yd ( c ) ˜ yd ( f ) / / ˜ yd ( c ′ ) yd ( d ) κ F ( d,z ) O O κ yd ( d,z ′ ) ; ; ✇✇✇✇✇✇✇✇✇ (which follows by the definition of ˜ yd ( f ) ) now immediately implies our thesis.Let us now define a function χ assigning to any natural transformation β : ˜ F → G a natural transformation χ ( β ) : F → G r . For any d ∈ D , wedefine χ ( β )( d ) : F ( d ) → G r ( d ) by setting, for each x ∈ F ( d ) , χ ( β )( x ) equalto the natural transformation β ◦ ˜ a x : ˜ yd → G , where a x : yd → F is thenatural transformation corresponding, via the Yoneda lemma, to the element x ∈ F ( c ) .To verify that χ ( β ) is well-defined, we have to check that for every arrow g : d → d ′ in D , the diagram F ( d ) F ( g ) (cid:15) (cid:15) χ ( β )( d ) / / G r ( d ) G r ( g )= −◦ ˜ yd (cid:15) (cid:15) F ( d ′ ) χ ( β )( d ′ ) / / G r ( d ′ ) commutes, i.e. for every x ∈ F ( d ) , G r ( g )( χ ( β )( d )( x )) = χ ( β )( d ′ )( F ( g )( x )) .Now, G r ( g )( χ ( β )( d )( x )) = β ◦ ˜ α x ◦ ˜ yg , while χ ( β )( d ′ )( F ( g )( x )) = β ◦ ˜ α F ( g )( x ) ;but α F ( g )( x ) = α x ◦ y D g , from which our thesis follows.The proof of the fact that the correspondences τ and β are natural in F and G is straightforward and left to the reader.58o conclude the proof of the theorem, it thus remains to show that τ and χ are inverse to each other. The verification of the fact that the unit andcounit of the adjunction coincide with those given in the statement of thetheorem is straightforward and left to the reader.Let us show that for any natural transformation α : F → G r , χ ( τ ( α )) = α . Let us set β = τ ( α ) . We have to prove that for any d ∈ D , x ∈ F ( d ) and z ∈ P ( c )( d ′ ) , α ( d )( x )( c ) ◦ κ y D d ( d ′ ,z ) = χ ( β )( d )( x )( c ) ◦ κ y D d ( d ′ ,z ) as functions y D d ( d ′ ) → G ( c ) , i.e. that for any element g ∈ y D d ( d ′ ) , α ( d )( x )( c )( κ y D d ( d ′ ,z ) ( g )) =( χ ( β )( d )( x )( c ) ◦ κ y D d ( d ′ ,z ) )( g ) . Now, by definition of the functor ˜( − ) and of thecorrespondence τ , the diagram ˜ y D d ( c ) ˜ a x ( c ) / / ˜ F ( c ) β ( c ) / / G ( c ) y D d ( d ′ ) κ y D d ( d ′ ,z ) O O a x ( d ′ ) / / F ( d ′ ) κ F ( d ′ ,z ) O O κ F ( d ′ ,z ) O O u ( d ′ ,z ) < < ①①①①①①①①① commutes, and since χ ( β )( x ) = β ◦ ˜ a x : ˜ y D d → G , we have that ( χ ( β )( d )( x )( c ) ◦ κ y D d ( d ′ ,z ) )( g ) is equal to α ( d ′ )( F ( g )( x ))( c )( κ y D d ′ ( d ′ ,z ) (1 d ′ )) .Now, the naturality diagram for α with respect to the arrow g : d → d ′ yields the equality α ( d )( x ) ◦ ˜ y D g = α ( d ′ )( F g ( x )) and hence the equality α ( d )( x )( c ) ◦ ˜ y D g ( c ) ◦ κ y D d ′ ( d ′ ,z ) = α ( d ′ )( F g ( x ))( c ) ◦ κ y D d ′ ( d ′ ,z ) .But the commutativity of the diagram ˜ y D d ′ ( c ) ˜ y D d ( g )( c ) / / ˜ y D d ( c ′ ) y D d ′ ( d ′ ) κ y D d ′ ( d ′ ,z ) O O y D g ( d ′ ) / / y D d ( d ′ ) κ y D d ( d ′ ,z ) O O (which follows by definition of the functor ˜( − ) ) implies that α ( d )( x )( c ) ◦ ˜ y D g ( c ) ◦ κ y D d ′ ( d ′ ,z ) = α ( d )( x )( c ) ◦ κ y D d ( d ′ ,z ) ◦ y D g . Therefore α ( d )( x )( c ) ◦ κ y D d ( d ′ ,z ) ◦ y D g = α ( d ′ )( F g ( x ))( c ) ◦ κ y D d ′ ( d ′ ,z ) and evaluating at d ′ yields the desired equality.Finally, let us show that the composite τ ◦ χ is equal to the identity.Let β : ˜ F → G be a natural transformation. We have to show that β = τ ( χ ( β )) . Let us set α = χ ( β ) . To prove that β = τ ( α ) it is equivalentto verify that for every object c ∈ C , any pair ( d, z ) with d ∈ D and z ∈ P ( c )( d ) and any element x ∈ F ( d ) , β ( c )( κ F ( d,z ) ( x )) = τ ( α )( c )( κ F ( d,z ) ( x )) .But τ ( α )( c )( κ F ( d,z ) ( x )) = α ( d )( x )( c )( κ y D d ( d,z ) (1 d )) = χ ( β )( x )( c )( κ y D d ( d,z ) (1 d )) =( β ◦ ˜ a x )( c )( κ y D d ( d,z ) (1 d )) = β ( c )(( ˜ a x )( c )( κ y D d ( d,z ) (1 d ))) . Thus our thesis follows59rom the commutativity of the diagram ˜ y D d ( c ) ˜ a x ( c ) / / ˜ F ( c ) yd ( d ) κ y D d ( d,z ) O O a x ( d ) / / F ( d ) , κ F ( d,z ) O O which is an immediate consequence of the definition of the functor ˜( − ) . (cid:3) Remarks 3.13. (a) Let f : D → C be a functor between two small cate-gories, and let P : C → [ D op , Set ] be the functor y C ( − ) ◦ f op . Noticethat P is the flat functor corresponding, via Diaconescu’s equivalence,to the essential geometric morphism [ D op , Set ] → [ C op , Set ] induced bythe functor f op : D op → C op . The functor ˜( − ) : [ D , E ] → [ C , E ] coin-cides with the left Kan extension functor along f , while the right adjointfunctor ( − ) r coincides with the functor − ◦ f : [ C , E ] → [ D , E ] .(b) Let u : Sh ( D , K ) → Sh ( C , J ) be a geometric morphism. Then for anyGrothendieck topos E , u induces as in section 3 a functor ξ E : Flat K ( D , E ) → Flat J ( C , E ) . By Diaconescu’s equivalence, the morphism u corresponds to a flat func-tor C → Sh ( D , K ) , which composed with the canonical geometric inclu-sion Sh ( D , K ) ֒ → [ D op , Set ] yields a functor P : C → [ D op , Set ] . Then ξ E coincides with the restriction of the functor ˜( − ) : [ D , E ] → [ C , E ] in-duced by P as in the theorem to the full subcategories Flat K ( D , E ) ֒ → [ D , E ] and Flat J ( C , E ) ֒ → [ C , E ] . In general, the right adjoint functor ( − ) r : [ C , E ] → [ D , E ] does not restrict to these subcategories, but ifit does, it becomes a right adjoint to the functor ξ E : Flat K ( D , E ) → Flat J ( C , E ) .This notably applies in the case of the geometric morphism p K : [ K , Set ] → Sh ( C T , J T ) considered in section 3.3. In this case, C = C T , D = K op and P is thefunctor C T → [ K , Set ] sending any geometric formula φ ( ~x ) to the functor M → [[ ~x . φ ]] M .(c) By a basic property of adjoint functors, the (left adjoint) functor ˜( − ) : [ D , E ] → [ C , E ] is full and faithful if and only if the unit η F : F → ( ˜ F ) r is an isomorphismfor any F in [ D , E ] . It easily follows (by purely formal considerations)60hat for every full subcategory H of [ D , E ] with canonical embedding i : H ֒ → [ D , E ] , the composite functor ˜( − ) ◦ i is full and faithful if andonly if the unit η F : F → ( ˜ F ) r is an isomorphism for any F in H .The following proposition will be useful in the sequel; see section 2.2 forthe notation employed in it. Proposition 3.14.
Let f : D → C be a functor between small categories, E a Grothendieck topos and F a functor C op → E . Then the E -indexed functor R ( F ◦ f op ) E → R F E sending any object ( c, x ) of R ( F ◦ f op ) E to the object ( f ( c ) , x ) of ( R F ) E is E -final if and only if the E -indexed functor R ǫ F E : R ( ˜ F ◦ f op ) E → R F E induced by the natural transformation ǫ F : ˜ F ◦ f → F of Theorem 3.12 is E -final. Proof
From the general analysis of section 3.1 we know that for any c ∈ C , ˜ F ◦ f op ( c ) = colim ( F ◦ f op ◦ π c ) , where π c is the canonical projection to D op from the category A c whose objects are the pairs ( d, h ) , where d isan object of the category D and h is an arrow c → f ( d ) in C , and whosearrows are the obvious ones. The arrow ǫ F : ˜ F ◦ f → F is defined bythe property that for any object ( d, h ) of A c , ǫ F ( c ) ◦ κ F ( d,h ) = F ( h ) , where κ F ( d,h ) : F ( f ( d )) → colim ( F ◦ f op ◦ π c ) is the canonical colimit arrow.Now, the thesis follows immediately from the fact that for any object ofthe category R ( ˜ F ◦ f op ) E of the form ( c, κ F ( d,h ) ◦ y ) , where y is a generalizedelement E → F ( f ( d )) , ( R ǫ F ) E (( c, κ F ( d,h ) ◦ y )) = ( c, F ( h ) ◦ y ) , invoking thefact that the colimit arrows are jointly epimorphic. (cid:3) In this section we establish some results on theories of presheaf type whichwill be important in the sequel.
For a theory of presheaf type T , it is possible to give an explicit syntacticdescription of the finitely presentable T -models; specifically, we have thefollowing result. Theorem 4.1.
Let T be a theory of presheaf type over a signature Σ and { ~x . φ } be a formula over Σ presenting a T -model U { ~x.φ } . Then U { ~x.φ } isisomorphic to the Σ -structure M { ~x.φ } defined as follows:(i) for any sort A over Σ , M { ~x.φ } A is equal to the set Hom C T ( { ~x . φ } , { x A . ⊤} ) of T -provably functional geometric formulae from { ~x . φ } to { x A . ⊤} ; ii) for any function symbol f : A , . . . , A n → B over Σ , the function M { ~x.φ } f : Hom C T ( { ~x . φ } , { x A . ⊤} ) × · · · × Hom C T ( { ~x . φ } , { x A n . ⊤} ) ∼ = Hom C T ( { ~x . φ } , { x A , . . . , x A n . ⊤} ) → Hom C T ( { ~x . φ } , { x B . ⊤} ) is equal to [ f ] ◦ − (where [ f ] : { x A , . . . , x A n . ⊤} → { x B . ⊤} isthe morphism in C T corresponding to f );(iii) for any relation symbol R A , . . . , A n , M { ~x.φ } R is the subobject of Hom C T ( { ~x . φ } , { x A . ⊤} ) × · · · × Hom C T ( { ~x . φ } , { x A n . ⊤} ) ∼ = Hom C T ( { ~x . φ } , { x A , . . . , x A n . ⊤} ) given by [ R ] ◦ − , (where [ R ] : { x A , . . . , x A n . R } { x A , . . . , x A n . ⊤} is the subobject in C T corre-sponding to R ). Proof
Recall that we have a canonical equivalence of categories
Flat J T ( C T , Set ) ≃ T -mod ( Set ) , sending any flat J T -continuous functor F : C T → Set to the T -model F ( M T ) ,where M T is the universal model of T in C T .We know from [8] that for any theory of presheaf type T over a signature Σ , any formula-in-context { ~x . φ } over Σ which presents a T -model is T -irreducible, in the sense that every J T -covering sieve on { ~x . φ } in C T ismaximal. From this it easily follows that the (flat) representable functor Hom C T ( { ~x . φ } , − ) : C T → Set is J T -continuous; indeed, for any J T -coveringsieve S on an object { ~y . ψ } of C T , any arrow γ : { ~x . φ } → { ~y . ψ } in C T factors through one of the arrows belonging to S as the pullback of S along γ is J T -covering and hence maximal. The image Hom C T ( { ~x . φ } , M T ) underthis functor of the model M T , which clearly coincides with the Σ -structure M { ~x.φ } in the statement of the theorem, is therefore a T -model. In order todeduce our thesis, it thus remains to verify that the model M { ~x.φ } satisfiesthe universal property of the T -model presented by the formula { ~x . φ } ,i.e. that for any T -model N in Set , the T -model homomorphisms M { ~x.φ } = Hom C T ( { ~x . φ } , M T ) → N are in natural bijection with the elements of theset [[ ~x . φ ]] N . But the T -model homomorphisms Hom C T ( { ~x . φ } , M T ) → N ,are in natural bijection, by the equivalence Flat J T ( C T , Set ) ≃ T -mod ( Set ) ,with the natural transformations Hom C T ( { ~x . φ } , − ) → F N , that is, by theYoneda lemma, with the elements of the set F N ( { ~x . φ } ) = [[ ~x . φ ]] N , asrequired. (cid:3) Remarks 4.2. (a) If T is a universal Horn theory (in the sense of [3])and φ ( ~x ) is a finite conjunction of atomic formulas in a context ~x =( x A , . . . , x A n ) then the set M { ~x.φ } A can be identified with the set ofequivalence classes of terms over Σ of type A , . . . , A n → A modulo theequivalence relation which identifies two terms t and t precisely whenthe sequent ( φ ⊢ ~x t = t ) is provable in T . Indeed, it is shown in [3] (cf.p. 120 therein) that any T -provably functional geometric formula θ ( ~x, ~y ) Σ is T -provably equivalent to a formula ofthe form ~y = ~t ( ~x ) , where ~t is a sequence of terms of the appropriate sortsin the context ~x .(b) Let T be a geometric theory over a signature Σ , K a small category ofset-based T -models and φ ( ~x ) a geometric formula over Σ presenting a T -model in K . If the geometric morphism p : [ K , Set ] → Sh ( C T , J T ) has the property that its inverse image p ∗ is full and faithful (for instance,if p is hyperconnected) then φ ( ~x ) is T -irreducible and the argument inthe proof of Theorem 4.1 applies yielding a syntactic description of themodel presented by φ ( ~x ) as specified in the statement of the theorem.Indeed, denoted by y and y ′ the Yoneda embeddings respectively of C T into Sh ( C T , J T ) and of K op into [ K , Set ] , we have that p ∗ ( y { ~x . φ } ) ∼ = y ′ ( M { ~x.φ } ) . Now, as y ′ ( M { ~x.φ } ) is an irreducible object in the topos [ K , Set ] and the property of an object of a topos to be irreducible isreflected by full and faithful inverse images of geometric morphisms, y { ~x . φ } is an irreducible object of the topos Sh ( C T , J T ) , equivalently φ ( ~x ) is T -irreducible, as required. For any first-order signature Σ and any Grothendieck topos E , we have a E -indexed category Σ - str ( E ) whose fibre at E ∈ E is the category Σ - str ( E ) and whose ‘change of base’ functors are the obvious pullback functors. Forany sort A over Σ , we have a E -indexed forgetful functor U A : Σ - str ( E ) → E E assigning to any Σ -structure M in the topos E /E the object M A . It is easyto see, by adapting the classical proof in the case E = Set and exploitingTheorem 2.12, that the E -indexed functors U A jointly create colimits of E -indexed diagrams defined on E -final and E -filtered subcategories A E of asmall E -indexed category. Indeed, the notion of Σ -structure only involvesfinite set-indexed limits, and, as we have remarked above, the colimit functor colim E : [ A E , E E ] → E E preserves them. Moreover, the structure used ininterpreting geometric formulae over Σ is all derived from set-indexed finitelimits and arbitrary colimits, which are both preserved by colim E (the factthat colimits commute with colimits is obvious, while the commutation withfinite limits has been observed above). This implies that for any geometrictheory T over Σ , the E -indexed full subcategory T -mod ( E ) of Σ - str ( E ) isclosed in Σ - str ( E ) under E -indexed colimits of diagrams defined on E -finaland E -filtered subcategories A E of a small E -indexed category.The following result asserts that the indexed category of models of ageometric theory in a Grothendieck topos is locally small.63 heorem 4.3. Let T be a geometric theory. Then for any T -models M and N in a Grothendieck topos E there exists an object Hom E T -mod ( E ) ( M, N ) of E , called the ‘object of T -model homomorphisms from M to N ’, satisfyingthe universal property that for any object E of E the generalized elements E → Hom E T -mod ( E ) ( M, N ) are in bijective correspondence, naturally in E ∈E , with the T -model homomorphisms ! ∗ E ( M ) → ! ∗ E ( N ) in T -mod ( E /E ) . Proof
We recall from [5] that for any small category C and any Grothendiecktopos E , for any two functors F, G : C → E there exists an object
Hom E ( F, G ) satisfying the property that for any object E of E the generalized elements E → Hom E ( F, G ) are in bijective correspondence, naturally in E ∈ E , withthe arrows ! ∗ E ◦ F → ! ∗ E ◦ G in [ C , E /E ] , that is with the natural transformations ! ∗ E ◦ F → ! ∗ E ◦ G . This implies that for any Grothendieck topos E the indexedcategory [ C , E ] E of functors on C with values in E and natural transformationsbetween them is locally small. Clearly, it follows at once that any E -indexedfull subcategory of [ C , E ] E , such as for example the indexed category of J -continuous flat functors on C with values in E , is also locally small.Now, every geometric theory T is Morita-equivalent to the theory of flat J T -continuous functors C T (cf. [21]); in other words, the E -indexed category T -mod ( E ) is equivalent to the E -indexed category of flat J T -continuous func-tors on C T with values in E . Hence T -mod ( E ) is locally small, as required. (cid:3) Remarks 4.4. (a) The assignment ( M, N ) → Hom E T -mod ( E ) ( M, N ) is func-torial both in M and N ; that is, any homomorphism f : M → M ′ in T -mod ( E ) (resp. any homomorphism g : N ′ → N in T -mod ( E ) ) inducesan arrow Hom E T -mod ( E ) ( f, N ) : Hom E T -mod ( E ) ( M ′ , N ) → Hom E T -mod ( E ) ( M, N ) (resp. an arrow Hom E T -mod ( E ) ( M, g ) :
Hom E T -mod ( E ) ( M, N ′ ) → Hom E T -mod ( E ) ( M, N )) functorially in f (resp. functorially in g ).(b) For any Grothendieck topos E and T -models M and N in E , we have acanonical embedding Hom F T -mod ( E ) ( M, N ) → Y A sort of Σ N A MA induced by arrows π A : Hom E T -mod ( E ) ( M, N ) → N A MA A of Σ ) defined in terms of generalized elements as follows: πA sends any arrow E → Hom E T -mod ( E ) ( M, N ) in E , corresponding toa T -model homomorphism r :! ∗ E ( M ) → ! ∗ E M in E /E , to the arrow E → N A MA whose transpose is the ‘evaluation’ ! ∗ E ( M A ) → ! ∗ E ( N A ) at A of r .(c) For any T -models M and N in a Grothendieck topos E and any geometricmorphism f : F → E , there is a canonical arrow f ∗ ( Hom E T -mod ( E ) ( M, N )) → Hom F T -mod ( F ) ( f ∗ ( M ) , f ∗ ( N )) . Indeed, this arrow corresponds, by the universal property of the object
Hom F T -mod ( F ) ( f ∗ ( M ) , f ∗ ( N )) to the T -model homomorphism f ∗ ( Hom E T -mod ( E ) ( M, N )) × f ∗ ( M ) → f ∗ ( Hom E T -mod ( E ) ( M, N )) × f ∗ ( N ) . in the topos F / ( f ∗ ( Hom F T -mod ( E ) ( M, N ))) whose first component, at anysort A , is the canonical projection and whose second component at A isthe arrow f ∗ ( Hom E T -mod ( E ) ( M, N )) × f ∗ ( M A ) ∼ = f ∗ ( Hom E T -mod ( E ) ( M, N ) × M A ) → f ∗ ( N A ) . obtained by taking the image under f ∗ of the arrow Hom E T -mod ( E ) ( M, N ) × M A → N A given by the transpose of the arrow π A : Hom E T -mod ( E ) ( M, N ) → N A MA defined above.(d) Since the E -indexed category T -mod ( E ) is locally small (cf. the proof ofTheorem 4.3), we have a E -indexed hom functor Hom E T -mod ( E ) ( − , − ) : T -mod ( E ) × T -mod ( E ) → E E The following proposition provides an explicit description of the gener-alized elements of the objects of homomorphism of Theorem 4.3 in a par-ticular case of interest. In the statement of the proposition, for a T -model M in a Grothendieck topos E , we denote by Hom E ( E, M ) the Σ -structurein Set obtained as the image of M under the product-preserving functor Hom E ( E, − ) : E →
Set . Proposition 4.5.
Let T be a geometric theory over a signature Σ , M a modelof T in a Grothendieck topos E , c a set-based T -model and E an object of E .Then the generalized elements x : E → Hom E T -mod ( E ) ( γ ∗E ( c ) , M ) correspondbijectively to the Σ -structure homomorphisms ξ x : c → Hom E ( E, M ) . roof By definition of
Hom E T -mod m ( E ) ( γ ∗E ( c ) , M ) , a generalized element E → Hom E T -mod m ( E ) ( γ ∗E ( c ) , M ) corresponds precisely to a T -model homo-morphism γ ∗E /E ( c ) → ! ∗ E ( M ) in the topos E /E . Concretely, such a T -modelhomomorphism consists of a family of arros τ A : γ ∗E /E ( cA ) → ! ∗ E ( M A ) in E /E indexed by the sorts A over Σ which satisfies the preservation con-ditions defining the notion of Σ -structure homomorphism. Now, each ofthe arrows τ A : γ ∗E /E ( cA ) → ! ∗ E ( M A ) corresponds, via the adjunction be-tween γ ∗E /E and the global section functor on the topos E /E , to a func-tion ξ A : cA → Hom E ( E, M A ) , and it is immediate to see that the above-mentioned preservation conditions translate precisely into the requirementthat the arrows ξ A : cA → Hom E ( E, M ) should yield a Σ -structure homo-morphism c → Hom E ( c, M ) . (cid:3) Remark 4.6.
If the model c is finitely presentable also as a T c -model, where T c is the cartesianization of T as defined in section 6.4 (notice that this isalways the case if Σ is finite and T has only a finite number of axioms,cf. Theorem 6.4 [6]) then, if φ ( ~x ) is a formula over Σ which presents it,the Σ -structure homomorphisms c → Hom E ( E, M ) can be identified withthe elements of the interpretation of the formula φ ( ~x ) in the Σ -structure Hom E ( E, M ) . We shall show in this section that the finitely presentable models of a theory T of presheaf type enjoy a strong form of finite presentability by a geometricformula with respect to the models of T in arbitrary Grothendieck toposes.Let T be a geometric theory over a signature Σ , c a set-based model of T , φ ( ~x ) a geometric formula over Σ and ~u an element of [[ ~x . φ ]] c . Then forany model M of T in a Grothendieck topos E there is a canonical arrow τ Mφ ( ~x ) ,~u : Hom E T -mod ( E ) ( γ ∗E ( c ) , M ) → [[ ~x . φ ]] M in E , defined on generalized elements as follows: τ Mφ ( ~x ) ,~u sends a generalizedelement E → Hom E T -mod ( E ) ( γ ∗E ( c ) , M ) , corresponding under the bijection of Proposition 4.5 to a Σ -structure ho-momorphism f : c → Hom E ( E, M ) , to the image of ~u under f ; noticethat such element indeed belongs to Hom E ( E, [[ ~x . φ ]] M ) since, as f is a Σ -structure homomorphism, the image of [[ ~x . φ ]] c under f is containedin [[ ~x . φ ]] Hom E ( E,M ) , which is contained in Hom E ( E, [[ ~x . φ ]] M ) since thefunctor Hom E ( E, − ) is cartesian. 66 efinition 4.7. Let T be a geometric theory over a signature Σ , and c amodel of T in Set . The T -model c is said to be strongly finitely presented ifthere exists a geometric formula φ ( ~x ) over Σ and a finite string of elements ~u of [[ ~x . φ ]] c , called the strong generators of c , such that for any T -model M in a Grothendieck topos E the arrow τ Mφ ( ~x ) ,~u : Hom E T -mod ( E ) ( γ ∗E ( M { ~x.φ } ) , M ) → [[ ~x . φ ]] M is an isomorphism (equivalently, the Σ -structure homomorphisms ξ : c → Hom E ( E, M ) are in natural bijection with the generalized elements E → [[ ~x . φ ]] M via the assignment ξ → ξ ( ~u ) ). Remark 4.8.
If the latter condition in the definition is satisfied by all models M of T inside Grothendieck toposes for E = 1 E then it is true in general, bythe localization principle. Theorem 4.9.
Let T be a theory of presheaf type classified by the topos [ K , Set ] , where K is a small subcategory of T -mod ( Set ) , M a T -model in aGrothendieck topos E and c a T -model in K . Then there is a natural bijectivecorrespondence between the Σ -structure homomorphisms c → Hom E ( E, M ) and the elements of the set Hom E ( E, F M c ) , where F M is the flat functor K op → E corresponding to the model M via the canonical Morita-equivalencefor T . Proof
We can clearly suppose without loss of generality E = 1 E . Theadjunction between γ ∗E and the global sections functor Γ E : E →
Set pro-vides a natural bijective correspondence between the elements of the set
Hom E ( E, F M c ) and the natural transformations γ ∗E ◦ yc → F M . By thecanonical Morita-equivalence τ E : Flat ( K op , E ) ≃ T -mod ( E ) , for T , such natural transformations are in natural bijective correspondencewith the T -model homomorphisms γ ∗E ( c ) ∼ = τ ( γ ∗E ◦ yc ) → τ ( F M ) = M . Butthese homomorphisms are, by Proposition 4.5, in natural bijective correspon-dence with the Σ -structure homomorphisms c → Hom E (1 E , M ) , as required. (cid:3) Corollary 4.10.
Let T be a theory of presheaf type and φ ( ~x ) a formulapresenting a T -model M { ~x.φ } . Then M { ~x.φ } is strongly finitely presented by φ ( ~x ) , i.e. for any T -model M in a Grothendieck topos E and any object E of E , the Σ -structure homomorphisms M { ~x.φ } → Hom E ( E, M ) correspondbijectively to the elements of the set Hom E ( E, [[ ~x . φ ]] M ) , via the assignmentsending a Σ -structure homomorphism M { ~x.φ } → Hom E ( E, M ) to the imageunder it of the generators of M { ~x.φ } . roof Clearly, we can suppose without loss of generality E = 1 E .By the results of [5], the canonical Morita-equivalence Flat ( f.p. T -mod ( Set ) op , E ) ≃ T -mod ( E ) for T can be described as the correspondence sending, on one hand, to any T -model M the flat functor F M := Hom T -mod ( E ) ( γ ∗E ( c ) , M ) and converselyto any flat functor F : f.p. T -mod ( Set ) op → E the model ˜ F ( M T ) , where ˜ F : C T → E is the extension of F to the syntactic category C T (in the sense ofsection 3.3). By Theorem 3.5, for any T -model M in a Grothendieck topos E ,there is an isomorphism z ( M, { ~x.φ } ) : F M ( M { ~x.φ } ) ∼ = ˜ F M ( { ~x . φ } ) = [[ ~x . φ ]] M .Thus, applying Theorem 4.9 to the model c = M { ~x.φ } , we obtain a bi-jective correspondence between the Σ -structure homomorphisms M { ~x.φ } → Hom E (1 E , M ) and the elements of the set Hom E (1 E , [[ ~x . φ ]] M ) . It remainsto show that this correspondence can be identified with the assignment send-ing a Σ -structure homomorphism M { ~x.φ } → Hom E (1 E , M ) to the image un-der it of the generators of M { ~x.φ } . Recall that the bijection of Theorem 4.9can be described as follows: any Σ -structure homomorphism f : M { ~x.φ } → Hom E (1 E , M ) , corresponding to a T -model homomorphism γ ∗E ( M { ~x.φ } ) → M , and hence, via the canonical Morita-equivalence for T , to a naturaltransformation γ ∗E ◦ yM { ~x.φ } ∼ = F γ ∗E ( M { ~x.φ } ) → F M (where y is the Yonedaembedding f.p. T -mod ( Set ) op ֒ → [ f.p. T -mod ( Set ) , Set ] ), is sent to the globalelement of F M ( M { ~x.φ } ) corresponding via the Yoneda lemma to this trans-formation. To deduce our thesis, it thus remains to verify that the canonicalisomorphism of functors γ ∗E ◦ yM { ~x.φ } = γ ∗E ◦ F M { ~x.φ } ∼ = F γ ∗E ( M { ~x.φ } ) , whenevaluated in M { ~x.φ } and composed with the isomorphism z ( γ ∗E ( M { ~x.φ } ) , { ~x.φ } ) : F γ ∗E ( M { ~x.φ } ) ( M { ~x.φ } ) ∼ = [[ ~x . φ ]] γ ∗E ( M { ~x.φ } ) ∼ = γ ∗E ([[ ~x . φ ]] M { ~x.φ } ) sends the co-product component of γ ∗E ( yM { ~x.φ } ( M { ~x.φ } )) corresponding to the identity on M { ~x.φ } to the coproduct component of γ ∗E ([[ ~x . φ ]] M { ~x.φ } ) corresponding tothe generators of M { ~x.φ } . By the naturality in E of the Morita-equivalencefor T we can clearly suppose, without loss of generality, E equal to Set .But from the proof of Theorem 4.1, it is clear that the generators of M { ~x.φ } correspond to the identity on { ~x . φ } ) via the Yoneda lemma; hence theycorrespond to the identity on M { ~x.φ } via the above-mentioned bijection, asrequired. Remark 4.11.
We can express the bijective correspondence of Corollary4.10 by saying that for any Grothendieck topos E the E -indexed functor [[ ~x . φ ]] − : T -mod ( E ) → E E assigning to any T -model M the interpretationof φ ( ~x ) in M is represented as a E -indexed functor by the object γ ∗E ( M { ~x.φ } ) ,in the sense of being naturally isomorphic to the E -representable functor Hom E T -mod ( E ) ( γ ∗E ( M { ~x.φ } ) , − ) . (cid:3) .4 Semantic E -finite presentability In this section, we introduce a semantic notion of E -finite presentabilityof a model M of a geometric theory T in a Grothendieck topos E , whichgeneralizes the classical notion in the context of finitely accessible categories,and show that all the ‘constant’ finitely presentable models of a theory ofpresheaf type in a Grothendieck topos E are E -finitely presentable.For any Grothendieck topos E , we can make any small category C intoa E -indexed category C E defined by: C E = C for all E ∈ E and C α = 1 C for all arrows α in E . To any (set-indexed) diagram D : C → E corre-sponds a E -indexed functor D E : C E → E E defined by: for any object E of E , D E E =! ∗ E ◦ D , where ! ∗ E : E → E /E is the pullback functor along theunique arrow E → E in E . Since the pullback functors preserve all smalllimits and colimits, giving a colimiting cocone (resp. a limiting cone) on thediagram D in the classical sense is equivalent to giving a E -indexed colim-iting cocone (resp. limiting cone) over the E -indexed diagram D E . Sincethe E -indexed category E E is complete, for any E -indexed category A E , the E -indexed functor category [ A E , E E ] is also complete (since limits are com-puted pointwise); in particular, it has limits of diagrams defined on E -indexedcategories of the form C E . On the other hand, if A E is a E -final E -filteredsubcategory of a small E -indexed category, the cocompleteness of E E as a E -indexed category ensures that there exists a well-defined E -indexed colimitfunctor colim E : [ A E , E E ] → E E . It is easy to see, by mimicking the classicalproof of the fact that finite limits commute with filtered colimits, that thisfunctor preserves limits of diagrams defined on E -indexed categories of theform C E for a finite category C . Definition 4.12.
Let T be a geometric theory and M be a model of T in a Grothendieck topos E . Then M is said to be E -finitely presentable ifthe E -indexed functor Hom E T -mod ( E ) ( M, − ) : T -mod ( E ) → E E of section 4.2preserves E -filtered colimits (of E -final and E -filtered subcategories of a small E -indexed category). Theorem 4.13.
Let T be a theory of presheaf type and c a finitely presentable T -model. Then for any Grothendieck topos E , the T -model γ ∗E ( c ) is E -finitelypresentable. Proof If c is presented by a geometric formula φ ( ~x ) over the signature Σ of T , by Corollary 4.10, the functor Hom E T -mod ( E ) ( γ ∗E ( c ) , M ) is naturallyisomorphic to the functor [[ ~x . φ ]] − : T -mod ( E ) → E E .This latter functor preserves E -filtered colimits (of E -final and E -filteredsubcategories of a small E -indexed category) since the structure used in in-terpreting geometric formulae over Σ is all derived from set-indexed finitelimits and arbitrary colimits, which, as remarked above, are all preserved by69 -indexed functors of the form colim E : [ A E , E E ] → E E where A E is a E -final E -filtered subcategory of a small E -indexed category. (cid:3) Remarks 4.14. (a) The proof of the theorem shows that, more generally,for any strongly finitely presentable model c of a geometric theory T in the sense of Definition 4.7 (for instance, a finite model of T if thesignature of T is finite - note that such a model is finitely presentedwith respect to the empty theory over the signature of T by Theorem6.4 [6]) and any Grothendieck topos E , the T -model γ ∗E ( c ) is E -finitelypresentable.(b) We could have more strongly required in Definition 4.12 the preserva-tion of all existing colimits of diagrams defined on E -filtered E -indexedcategories, in the sense of Definition 2.1. The theorem remains true withrespect to this stronger notion, but a smallness condition for the domaincategory, such as the requirement for it to be a E -final and E -filtered E -indexed subcategory of a small E -indexed category, is necessary todispose of the explicit characterization of filtered colimits provided byCorollary 2.24.The following proposition provides an explicit characterization of the E -finitely presentable models of a geometric theory T . Proposition 4.15.
Let T be a geometric theory, E a Grothendieck toposand c a T -model in E . Then c is E -finitely presentable if and only if forevery E -indexed diagram D : A E → T -mod ( E ) defined on a E -filtered E -final subcategory A E of a small E -indexed category with E -indexed colimitingcocone ( M, µ ) (we denote by µ ( E,x ) : D E ( x ) → ! ∗ E ( M ) the colimit arrows),the following conditions are verified:(a) For any object E of E and T -model homomorphism h :! ∗ E ( c ) → ! ∗ E ( M ) inthe topos E /E there exists an epimorphic family { e i : E i → E | i ∈ I } in E and for each i ∈ I an object x i of A E i and a T -model homomorphism α i :! ∗ E i ( c ) → D E i ( x i ) in the topos E /E i such that for all i ∈ I , µ ( E i ,x i ) ◦ α i = e ∗ i ( h ) ;(b) For any pairs ( x, y ) and ( x ′ , y ′ ) , where x and x ′ are objects of A E , f is anarrow F → E in E , y is a T -model homomorphism ! ∗ F ( c ) → f ∗ ( D E ( x )) in E /F and y ′ is a T -model homomorphism ! ∗ F ( c ) → f ∗ ( D E ( x ′ )) in E /F ,we have f ∗ ( µ E ( x )) ◦ y = f ∗ ( µ E ( x ′ )) ◦ y ′ if and only if there exists anepimorphic family { f i : F i → F | i ∈ I } in E and for each i ∈ I arrows g i : A f ◦ f i ( x ) → z i and h i : A f ◦ f i ( x ′ ) → z i in the category A F i such that D F i ( g i ) ◦ f ∗ i ( y ) = D F i ( h i ) ◦ f ∗ i ( y ′ ) . Proof
The proposition follows as an an immediate consequence of Corollary2.24, applied to the composite E -indexed functor Hom E T -mod ( E ) ( c, − ) ◦ D . (cid:3) emark 4.16. By the construction of (small) E -indexed colimits in the E -indexed category T -mod ( E ) , a E -indexed cocone ( M, µ ) over a diagram D : A E → T -mod ( E ) as in the statement of the proposition is colimitingif and only if for every sort A over the signature of T , U A (( M, µ )) is acolimiting cocone over the diagram U A ◦ D in E E . In this section we establish our main characterization theorem providingnecessary and sufficient conditions for a geometric theory to be of presheaftype. These conditions are entirely expressed in terms of the models of thetheory in arbitrary Grothendieck toposes.We shall first prove the theorem and then proceed to reformulate itsconditions in more concrete terms so for them to be directly applicable inpractice.
Recall from [5] that the classifying topos of a theory of presheaf type T canbe canonically represented as the topos [ f.p. T -mod ( Set ) , Set ] of set-valuedfunctors on the category f.p. T -mod ( Set ) of finitely presentable T -models in Set . This is not the only possible representation of the classifying topos of T as a presheaf topos, but for any small category K , T is classified by thetopos [ K , Set ] if and only if the Cauchy-completion of K is equivalent tof.p. T -mod ( Set ) . Theorem 5.1.
Let T be a geometric theory over a signature Σ and let K be a full subcategory of f.p. T -mod ( Set ) . Then T is a theory of presheaf typeclassified by the topos [ K , Set ] if and only if all of the following conditionsare satisfied:(i) For any T -model M in a Grothendieck topos E , the functor H M := Hom E T -mod ( E ) ( γ ∗E ( − ) , M ) : K op → E is flat;(ii) The extension ˜ H M : C T → E of the functor H M : K op → E to thesyntactic category C T (in the sense of section 3.3) satisfies the propertythat the canonical morphism ˜ H M ( M T ) → M is an isomorphism;(iii) Any of the following conditions (equivalent, under assumptions ( i ) and ( ii ) ) is satisfied:(a) For any model c in K , T -model M in a Grothendieck topos E andgeometric morphism f : F → E , the canonical morphism f ∗ ( Hom E T -mod ( E ) ( γ ∗E ( c ) , M )) → Hom F T -mod ( F ) ( γ ∗F ( c ) , f ∗ ( M )) rovided by Remark 4.4(c) using the identification γ ∗F ( c ) ∼ = f ∗ ( γ ∗E ( c )) ,is an isomorphism;(b) For any flat functor F : K op → E , the canonical natural transfor-mation η F : F → Hom E T -mod ( E ) ( γ ∗E ( − ) , ˜ F ( M T )) ∼ = Hom E Flat J T ( C T , E ) ( γ ∗E ◦ ˜ y K ( − ) , ˜ F ) of Theorem 3.12 is an isomorphism, where y K : K op ֒ → Flat ( K op , Set ) is the Yoneda embedding;(c) The functor u T ( K , E ) : Flat ( K op , E ) → Flat J T ( C T , E ) ≃ T -mod ( E ) of section 3.3 is full and faithful. Proof
Let us begin by proving that each of the three listed conditions arenecessary.By the results in [5], if T is of presheaf type classified by the topos [ K , Set ] then we have a Morita-equivalence τ E : Flat ( K op , E ) ≃ T -mod ( E ) which can be supposed canonical without loss of generality, i.e. which sendsany finitely presentable T -model c in K to the functor γ ∗E ◦ y K c . It follows thatfor any T -model M in a Grothendieck topos E , with corresponding flat func-tor F M under this Morita-equivalence, the object Hom E T -mod ( E ) ( γ ∗E ( c ) , M ) isisomorphic to the object Hom E Flat ( K op , E ) ( γ ∗E ◦ y K c, F M ) ∼ = F M ( c ) , naturallyin M and in c . Therefore the functor Hom E T -mod ( E ) ( γ ∗E ( − ) , M ) : K op → E is flat, it being isomorphic to F M . This proves that condition ( i ) of thetheorem is satisfied.Next, we notice that the left-to-right functor forming the canonical Morita-equivalence τ E for T can be described as follows: for any flat functor F : K op → E , the T -model corresponding to it is naturally isomorphic to themodel ˜ F ( M T ) . Indeed, as the Morita-equivalence for T is canonical, it isinduced by the canonical geometric morphism (in fact, equivalence) p K :[ K , Set ] → Sh ( C T , J T ) , i.e. it is given by the composite of the inducedequivalence Flat ( K op , E ) ≃ Flat J T ( C T , E ) with the canonical equivalence Flat J T ( C T , E ) ≃ T -mod ( E ) sending any flat J T -continuous functor G on C T to the T -model G ( M T ) .The fact that τ E is an equivalence thus implies that the canonical arrow ˜ H M → M is an isomorphism. This shows that condition ( ii ) of the theoremis satisfied. 72he fact that condition ( iii )( c ) is satisfied also immediately follows fromthe fact that τ E is an equivalence.We have thus proved that conditions ( i ) , ( ii ) and ( iii )( c ) of the theoremare necessary.Let us now show that conditions ( i ) , ( ii ) and ( iii )( b ) are, all together,sufficient for the theory T to be classified by the topos [ K , Set ] .First, we notice that condition ( iii )( b ) implies condition ( iii )( a ) underthe assumption that condition ( i ) holds. Indeed, for any T -model M in aGrothendieck topos E , condition ( i ) ensures that the functor H M : K op → E is flat. On the other hand, for any geometric morphism f : F → E condition ( iii )( b ) yields, in view of the naturality in E of the operation ˜( − ) , a naturalisomorphism between the flat functor f ∗ ◦ H M and the functor H f ∗ ( M ) . Thisensures that the requirement of condition ( iii )( a ) is satisfied.Under conditions ( i ) , ( ii ) and ( iii )( b ) , we shall construct, for any Grothen-dieck topos E , a categorical equivalence Flat ( K op , E ) ≃ T -mod ( E ) natural in E .We shall define two functors G E : Flat ( K op , E ) → T -mod ( E ) and H E : T -mod ( E ) → Flat ( K op , E ) which are natural in E and categorical inverses toeach other (up to isomorphism).We set H E ( M ) equal to the functor Hom E T -mod ( E ) ( γ ∗E ( − ) , M ) : K op → E . This assignment is natural in M (cf. Remark 4.4) and hence defines a functor H E : T -mod ( E ) → Flat ( K op , E ) , which is natural in E by condition ( iii )( a ) .In the converse direction, for any flat functor F : K op → E we set G E ( F ) = ˜ F ( M T ) . Clearly, this assignment is natural in F and defines afunctor G E : Flat ( K op , E ) → T -mod ( E ) .For each Grothendieck topos E , the functors G E and H E are categoricalinverses to each other (up to isomorphism). Indeed, condition ( iii )( b ) ensuresthat H E ◦ G E is naturally isomorphic to the identity, while condition ( ii ) ensures that G E ◦ H E is naturally isomorphic to the identity.Now, the functors G E : Flat ( K op , E ) → T -mod ( E ) and H E : T -mod ( E ) → Flat ( K op , E ) are natural in E and therefore induce geometric morphisms G : [ f.p. T -mod ( Set ) , Set ] → Sh ( C T , J T ) H : Sh ( C T , J T ) → [ K , Set ] . The fact that G and H are two halves of a categorical equivalence between [ K , Set ] and Sh ( C T , J T ) follows immediately from the fact that for everyGrothendieck topos E there are natural isomorphisms between H E ◦ G E andthe identity functor on Flat ( K op , E ) and between G E ◦ H E and the iden-tity functor on T -mod ( E ) , provided respectively by condition ( iii )( b ) andcondition ( ii ) .Finally, let us show that, under the assumption of conditions ( i ) and ( ii ) of the theorem, conditions ( iii )( a ) , ( iii )( b ) , ( iii )( c ) are all equivalent.Conditions ( iii )( b ) and ( iii )( c ) are equivalent by Remark 3.13(c). Thenecessity of condition ( iii )( a ) follows from Remark 4.11 and the fact that anyfinitely presentable model of a theory of presheaf type is finitely presented(cf. [8]).Having already verified that conditions ( i ) , ( ii ) and ( iii )( b ) imply alltogether that T is classified by the topos [ K , Set ] , and that conditions ( iii )( a ) and ( iii )( d ) are both necessary conditions for T to be of presheaf type, itremains to check that, under conditions ( i ) and ( ii ) of the theorem, condition ( iii )( a ) implies condition ( iii )( b ) . We shall do so by verifying that conditions ( i ) , ( ii ) and ( iii )( a ) imply all together that T is classified by the presheaftopos [ K , Set ] .Under conditions ( i ) , ( ii ) and ( iii )( a ) , we clearly have functors G E : Flat ( K op , E ) → T -mod ( E ) and H E : T -mod ( E ) → Flat ( K op , E ) which are natural in E and therefore induce geometric morphisms G : [ K , Set ] → Sh ( C T , J T ) and H : Sh ( C T , J T ) → [ K , Set ] . Notice that the morphism G coincides with the morphism p K canonicallyinduced by the universal property of the classifying topos for T by the T -models in K .Notice that for any T -model P in E , H E ( P ) = f ∗ P ◦ H ∗ ◦ y K , where f P : E → Sh ( C T , J T ) is the geometric morphism corresponding to P via theuniversal property of the classifying topos and y K : K op → [ K , Set ] is theYoneda embedding.Under our assumptions, we have to prove that H ◦ G ∼ = 1 [ K , Set ]
74r equivalently, that G ∗ ◦ H ∗ ◦ y K ∼ = y K .Let us consider the geometric morphisms e N : Set → [ K , Set ] corresponding to the models N of T in K . As the e N are jointly surjective,it is equivalent to prove that e ∗ N ◦ G ∗ ◦ H ∗ ◦ y K ∼ = e ∗ N ◦ y K = Hom K ( − , N ) naturally in N ∈ K .Let us denote by M G the T -model in [ K , Set ] corresponding to the geo-metric morphism G . Clearly, for any T -model N in K , e ∗ N ( M G ) ∼ = N .We have that G ∗ ◦ H ∗ ◦ y K = H [ K , Set ] ( M G ) . But H [ K , Set ] ( M G ) = Hom ( γ ∗ [ K , Set ] ( − ) , M G ) , and, by condition ( iii )( a ) , e ∗ N ( Hom ( γ ∗ [ K , Set ] ( − ) , M G )) ∼ = Hom T -mod ( Set ) ( − , e ∗ N ( M G )) ∼ = Hom K ( − , N ) , (we have omitted the subscripts and superscripts in the Hom s above tolighten the notation), as required.On the other hand, if condition ( ii ) holds then G ◦ H is isomorphic tothe identity, so we can conclude that T is classified by the topos [ K , Set ] .In particular, T satisfies condition ( iii )( b ) . This completes the proof of thetheorem. (cid:3) Remarks 5.2. (a) The following condition is sufficient, together with con-ditions ( i ) and ( ii ) of the theorem (or equivalently, together with con-dition ( i ) and the requirement that the models in K should be jointlyconservative for T ), to ensure that T is classified by the topos [ K , Set ] but necessary only if one assumes the axiom of choice:( ∗ ) There is an assignment M → φ M ( ~x M ) to a T -model in K of a formula φ M ( ~x M ) presenting it such that every T -model homomorphism M → N between models in K is induced by a T -provably functional formula from φ N ( ~x N ) to φ M ( ~x M ) .This condition can be easier to verify in practice than the original con-dition since it allows to work with distinguished presentations (ratherthan with all of them) of finitely presentable models of the theory.The necessity of condition ( ∗ ) , under the axiom of choice, was estab-lished in [8]. In the converse direction, it suffices by Theorem 5.1 toprove that, under conditions ( i ) and ( ii ) (or equivalently, under condi-tion ( i ) and the assertion that the models in K are jointly conservativefor T ), condition ( ∗ ) implies condition ( iii )( c ) . First, we remark thatunder either of these assumptions, the canonical geometric morphism p K : [ K , Set ] → Sh ( C T , J T ) is a surjection. From this it easily followsthat for any geometric formulae { ~x . φ } and { ~y . ψ } over Σ respectivelypresenting models c and d in K , there can be at most one provably func-tional formula { ~x . φ } → { ~y . ψ } over Σ , up to T -provable equivalence,75nducing a given homomorphism of T -models d → c . Condition ( ∗ ) thusimplies that we have a well-defined full and faithful functor K → C T .From this it is immediate to see, by invoking Theorem 3.5 and Remark3.6, that condition ( iii )( c ) is satisfied.(b) If conditions ( i ) and ( ii ) in the theorem are satisfied then the canonicalgeometric morphism p K : [ K , Set ] → Sh ( C T , J T ) is a surjection; in other words, the models in K are jointly conservativefor T . Indeed, by condition ( i ) the functor H M , where M is the universalmodel of T lying in its classifying topos, is flat. By applying Corollary3.7 to it we thus obtain that for any geometric sequent σ = ( φ ⊢ ~x ψ ) over the signature of T , ˜ H M ( { ~x . φ } ) ≤ ˜ H M ( { ~x . ψ } ) ; condition ( ii ) ,combined with the conservativity of M , thus allows to conclude that σ is provable in T , as required.(c) Under condition ( i ) , if condition ( iii )( a ) is satisfied and the models in K are jointly conservative for T (that is, every geometric sequent over Σ which is valid in every model in K is provable in T ) then T satisfiescondition ( ii ) . Indeed, the assertion that the models in K should beenough for the theory T is precisely equivalent to the requirement thatthe geometric morphism p K should be a surjection. By the naturality in E of the operation ˜( − ) and that of the functor H E (notice that the latterfollows from condition ( iii )( a ) ), it suffices to verify, since p K is surjective,that the canonical morphism ˜ H M ( M T ) → M is an isomorphism for M equal to a model in K . But the fact that this condition holds is obvious,as required.(d) Conditions ( ii ) and ( iii )( c ) in Theorem 5.1 admit invariant formulations,which can be profitably exploited in presence of different representationsfor the classifying topos of T . Indeed, they can both be entirely refor-mulated in terms of the extension of flat functors operation (in the senseof section 3.1) along the canonical geometric morphism p K : [ K , Set ] → Set [ T ] to the classifying topos Set [ T ] for T induced by the T -models in K .Specifically, any site of definition ( C , J ) for Set [ T ] gives rise to a functor G ( C ,J ) E : Flat ( K op , E ) → Flat J ( C , E ) . Condition ( iii )( c ) asserts that this functor, whose explicit description isgiven section 3.1, is full and faithful, while condition ( ii ) asserts that,denoting by u E : Flat J ( C , E ) ≃ T -mod ( E ) T , for any T -model M in a Grothendieck topos E ,the canonical morphism u E ( G ( C ,J ) E ( H M )) → M is an isomorphism.(e) Condition ( iii )( b ) can be split into two separate conditions: η F is point-wise monic and η F is pointwise epic. We shall refer to the first conditionas to condition ( iii )( b ) - (1) and to the second as to condition ( iii )( b ) - (2) .By Theorem 3.12 and Lemma 7.11 below, condition ( iii )( b ) - (1) is equiv-alent to the requirement that the functor u T ( K , E ) should be faithful. In this section we shall give ‘concrete’ reformulations of the conditions ofTheorem 5.1, in full generality as well as in some particular cases in whichthey admit relevant simplifications. ( i ) In this section we shall give an explicit reformulation of condition ( i ) inTheorem 5.1. Theorem 5.3.
Let T be a geometric theory, K a small category of set-basedmodels of T , E a Grothendieck topos with a separating set S and M a T -modelin E . Then the following conditions are equivalent:(i) The functor H M := Hom E T -mod ( E ) ( γ ∗E ( − ) , M ) : K op → E is flat.(ii) (a) There exists an epimorphic family { E i → E | i ∈ I, E i ∈ S } andfor each i ∈ I a T -model c i in K and a Σ -structure homomorphism c i → Hom E ( E i , M ) ;(b) For any T -models c and d in K and Σ -structure homomorphisms x : c → Hom E ( E, M ) (where E ∈ S ) and y : d → Hom E ( E, M ) ,there exists an epimorphic family { e i : E i → E | i ∈ I, E i ∈ S } and for each i ∈ I a T -model b i in K , T -model homomorphisms u i : c → b i , v i : d → b i and a Σ -structure homomorphism z i : b i → Hom E ( E i , M ) such that Hom E ( e i , M ) ◦ x = z i ◦ u i and Hom E ( e i , M ) ◦ y = z i ◦ v i .(c) For any two parallel homomorphisms u, v : d → c of T -models in K and any Σ -structure homomorphism x : c → Hom E ( E, M ) in E (where E ∈ S ) for which x ◦ u = x ◦ v , there is an epimorphicfamily { e i : E i → E | i ∈ I, E i ∈ S } in E and for each index i homomorphism of T -models in K w i : c → b i and a Σ -structurehomomorphism y i : b i → Hom E ( E i , M ) such that w i ◦ u = w i ◦ v and y i ◦ w i = Hom E ( e i , M ) ◦ x . Proof
This follows immediately from Proposition 4.5 in view of the charac-terization of flat functors as filtering functors established in chapter VII of[26] and reported in section 2.7. (cid:3)
Remarks 5.4. (a) If for any T -model M in a Grothendieck topos E andany object E of E the Σ -structure Hom E ( E, M ) is a T -model then thethree conditions ( a ) , ( b ) and ( c ) are satisfied if they are satisfied in thecase E = Set .(b) Condition ( ii )( b ) implies condition ( ii )( c ) if all the T -model homomor-phisms in any Grothendieck topos are monic.(c) Condition ( ii )( a ) follows from condition ( ii )( a ) of Theorem 5.7 belowif the signature of T contains at least one constant. Indeed, for any T -model M in a Grothendieck topos, the interpretation of such constantin M will be an arrow → M A in the topos, where A is the sort of theconstant; applying part ( a ) of condition ( ii ) of Theorem 5.7 thus yieldsan epimorphic family satisfying the required property.(d) If all the T -models in K are finitely generated as Σ -structures and allthe Σ -structure homomorphisms of the form c → Hom E ( E, M ) (where c is a T -model in K and M is a T -model in E ) are injective if E ≇ ,a sufficient condition for property ( ii )( b ) to hold is that, for any con-text ~x , the disjunction ( ⊤ ⊢ ~x W φ ( x ) ∈I ~x K φ ( ~x )) be provable in T , where I ~x K is the set of geometric formulae in the context ~x which stronglyfinitely present a T -model in K (in the sense of Definition 4.7). Indeed,for any two Σ -structure homomorphisms x : c → Hom E ( E, M ) and y : d → Hom E ( E, M ) , where c and d are finitely generated Σ -structuresand E ≇ , the substructure r e : e ֒ → Hom E ( E, M ) of Hom E ( E, M ) generated by the images of c under x and of y under y is finitely gen-erated, say by elements ξ , . . . , ξ n ; by choosing a context ~x of the samelength as the number of generators of e , we obtain an epimorphic family { e i : E i → E | i ∈ I, E i ∈ S, E i ≇ } with the property that for each i ∈ I there exists a geometric formula φ i ( ~x ) strongly presenting a T -model u i in K such that ( ξ ◦ e i , . . . , ξ n ◦ e i ) factors through [[ ~x . φ i ]] M .Therefore, by the universal property of u i as T -model strongly presentedby the formula φ i ( ~x ) , for each i ∈ I we have a Σ -structure homomor-phism z i : u i → Hom E ( E i , M ) which sends the generators of u i to theelement ( ξ ◦ e i , . . . , ξ n ◦ e i ) . Since z i is injective by our hypothesis andits image contains a set of generators for e , we have a factorization of78 om E ( e i , M ) ◦ r e through r i . Therefore both Hom E ( e i , M ) ◦ x and Hom E ( e i , M ) ◦ y factor through z i . This shows that condition ( ii )( b ) issatisfied.Let us suppose that every T -model in K is strongly finitely presented bya formula over Σ (in the sense of Definition 4.7) and that every T -modelhomomorphism between two models in K is induced by a T -provably func-tional formula between formulas which present them. Then, in light of thediscussion preceding Definition 4.7, we can express the conditions for thefunctor H M to be flat (equivalently, filtering) in terms of the satisfaction by M of certain geometric sequents involving these formulas. Specifically, wehave the following result. Theorem 5.5.
Let T be a geometric theory over a signature Σ , K a smallfull category of the category of set-based T -models and P a family of geo-metric formulae over Σ such that every T -model in K is strongly presentedby a formula in P and for any two formulae φ ( ~x ) and ψ ( ~y ) in P present-ing respectively models c and d in K , any T -model homomorphism d → c is induced by a T -provably functional formula from φ ( ~x ) to ψ ( ~y ) . Then T satisfies condition ( i ) of Theorem 5.1 with respect to the category K if andonly if the following conditions are satisfied:(i) The sequent ( ⊤ ⊢ [] _ φ ( ~x ) ∈P ( ∃ ~x ) φ ( ~x )) is valid in M ;(ii) For any formulae φ ( ~x ) and ψ ( ~y ) in P , the sequent φ ( ~x ) ∧ ψ ( ~y ) ⊢ ~x,~y _ χ ( ~z ) ∈P , { ~x.φ } θ ←{ ~z.χ } θ →{ ~y.ψ } in C T ( ∃ ~z )( θ ( ~z, ~x ) ∧ θ ( ~z, ~y )) , is valid in M ;(iii) For any T -provably functional formulae θ , θ : φ ( ~x ) → ψ ( ~y ) betweentwo formulae φ ( ~x ) and ψ ( ~y ) in P , the sequent θ ( ~x, ~y ) ∧ θ ( ~x, ~y ) ⊢ ~x,~y _ χ ( ~z ) ∈P , { ~z.χ } τ →{ ~x.φ } in C T ,τ ∧ θ ⊣⊢ T τ ∧ θ ( ∃ ~z ) τ ( ~z, ~x ) is valid in M . Proof
The fact that every model c in K is strongly finitely presented by aformula φ ( ~x ) in P ensures that for any object E of E and any T -model M in E Σ -homomorphism c → Hom E ( E, M ) are in bijective correspondence withthe generalized elements E → [[ ~x . φ ]] M in E . The thesis then follows fromthe Kripke-Joyal semantics for the topos E , noticing that by our hypothesesfor any two formulae φ ( ~x ) and ψ ( ~y ) in P presenting respectively models c and d in K , any T -model homomorphism d → c is induced by a T -provablyfunctional formula from φ ( ~x ) to ψ ( ~y ) . (cid:3) Remarks 5.6. (a) For any geometric theory T and category K satisfyingthe hypotheses of the theorem, all the sequents in the statement of thetheorem are satisfied by every model in K . Therefore, adding them to T yields a quotient of T satisfying condition ( i ) of Theorem 5.1 withrespect to the category K .(b) Under the alternative hypothesis that every model of K is both stronglyfinitely presentable and finitely generated (with respect to the same gen-erators), for any two formulae φ ( ~x ) and ψ ( ~y ) in P presenting respectivelymodels c and d in K , the T -model homomorphisms c → d are in bijection(via the evaluation of such homomorphisms at the generators of c ) withthe tuples of elements of d which satisfy φ , each of which has the form ( t ( ~z ) , . . . , t n ( ~z )) (where n is the length of the context ~x ) for some terms t , . . . , t n in the context ~z over the signature Σ . Conditions ( ii ) and ( iii ) of Theorem 5.5 can thus be reformulated more explicitly as follows: ( ii ′ ) For any formulae φ ( ~x ) and ψ ( ~y ) in P , where ~x = ( x A , . . . , x A n n ) and ~y = ( y B , . . . , y B m m ) , the sequent ( φ ( ~x ) ∧ ψ ( ~y ) ⊢ ~x,~y _ χ ( ~z ) ∈P ,t A ( ~z ) ,...,t Ann ( ~z ) s B ( ~z ) ,...,s Bmm ( ~z ) ( ∃ ~z )( χ ( ~z ) ∧ ^ i ∈{ ,...,n } ,j ∈{ ,...,m } ( x i = t i ( ~z ) ∧ y j = s j ( ~z )))) , where the disjunction is taken over all the formulae χ ( ~z ) in P and allthe sequences of terms t A ( ~z ) , . . . , t A n n ( ~z ) and s B ( ~z ) , . . . , s B m m ( ~z ) whoseoutput sorts are respectively A , . . . , A n , B , . . . , B m and such that, de-noting by ~ξ the set of generators of the model M { ~z.χ } (strongly) finitelypresented by the formula χ ( ~z ) , ( t A ( ~ξ ) , . . . , t A n n ( ~ξ )) ∈ [[ ~x . φ ]] M { ~z.χ } and ( s B ( ~ξ ) , . . . , s B m m ( ~ξ )) ∈ [[ ~y . ψ ]] M { ~z.χ } , is valid in M ; ( iii ′ ) For any formulae φ ( ~x ) and ψ ( ~y ) in P , where ~x = ( x A , . . . , x A n n ) and ~y = ( y B , . . . , y B m m ) , and any terms t A ( ~y ) , s A ( ~y ) , . . . , t A n n ( ~y ) , s A n n ( ~y ) whose output sorts are respectively A , . . . , A n , the sequent ( ^ i ∈{ ,...,n } ( t i ( ~y ) = s i ( ~y )) ∧ φ ( t /x , . . . , t n /x n ) ∧ φ ( s /x , . . . , s n /x n ) ∧ ψ ( ~y ) ⊢ ~y _ χ ( ~z ) ∈P ,u B ( ~z ) ,...,u Bmm ( ~z ) (( ∃ ~z )( χ ( ~z ) ∧ ^ j ∈{ ,...,m } ( y j = u j ( ~z ))) , χ ( ~z ) in P andall the sequences of terms u B ( ~z ) , . . . , u B m m ( ~z ) whose output sorts arerespectively B , . . . , B m and such that, denoting by ~ξ the set of gener-ators of the model M { ~z.χ } (strongly) finitely presented by the formula χ ( ~z ) , ( u B ( ~ξ ) , . . . , u B m m ( ~ξ )) ∈ [[ ~y . ψ ]] M { ~z.χ } and t i ( u ( ~ξ ) , . . . , u m ( ~ξ )) = s i ( u ( ~ξ ) , . . . , u m ( ~ξ )) in M { ~z.χ } for all i ∈ { , . . . , n } , is valid in M . ( ii ) In this section we shall give concrete reformulations of condition ( ii ) of Theo-rem 5.1, under the assumption that condition ( i ) of Theorem 5.1 is satisfied.First, notice that the canonical morphism ˜ H M ( M T ) → M considered incondition ( ii ) is an isomorphism if and only if for every sort A over Σ , theinduced arrow ˜ H M ( { x A . ⊤} ) → M A is an isomorphism in E . To understandthis condition more concretely, we apply Proposition 3.8 to the geometricformula { x A . ⊤} and the flat functor F = H M : K op → E . Recall that thecategory A K{ x A . ⊤} defined in that context has as objects the pairs ( c, z ) where c is a T -model in K and z is an element of the set cA and as arrows ( c, z ) → ( d, w ) the T -model homomorphisms g : d → c in K such that gA ( w ) = z ,and that the equivalence relation R { x A . ⊤} is defined by the condition that forany generalized elements x : E → F ( c ) and x ′ : E → F ( d ) , ( x, x ′ ) ∈ R { x A . ⊤} if and only if there exists an epimorphic family { e i : E i → E | i ∈ I } and foreach index i ∈ I a T -model a i in K , a generalized element h i : E i → F ( b i ) and two T -model homomorphisms f i : c → b i and f ′ i : d → b i in K such that f i A ( z ) = f ′ i A ( w ) and h F ( f i ) , F ( f ′ i ) i ◦ h i = h x, x ′ i ◦ e i .This yields that the canonical arrow ˜ H M ( { x A . ⊤} ) → M A is an isomor-phism if and only if (using the notation of Proposition 3.8):(1) the canonical arrows κ ( c,z ) : H M ( c ) → M A for ( c, z ) ∈ A { x A . ⊤} arejointly epimorphic and(2) for any two objects ( c, z ) and ( d, w ) of the category A { x A . ⊤} and anygeneralized elements x : E → H M ( c ) and x ′ : E → H M ( d ) (where E ∈ S ), κ ( c,z ) ◦ x = κ ( d,w ) ◦ x ′ if and only if ( x, x ′ ) ∈ R { x A . ⊤} .Thanks to the identification between the generalized elements x : E → H M ( c ) and the Σ -structure homomorphisms f x : c → Hom E ( E, M ) providedby Proposition 4.5, we can rewrite conditions (1) and (2) more explicitly.To this end, we preliminarily notice that for any object ( c, z ) of the cate-gory A { x A . ⊤} , the canonical arrow κ ( c,z ) : H M ( c ) = Hom E T -mod ( E ) ( γ ∗E ( c ) , M ) → M A can be described in terms of generalized elements as the arrow sendingany generalized element x : E → H M ( c ) , corresponding via the identificationof Proposition 4.5 to a Σ -structure homomorphism f x : c → Hom E ( E, M ) ,to the generalized element E → M A of M A given by f x A ( z ) . Therefore, for81ny generalized elements x : E → H M ( c ) and x ′ : E → H M ( d ) , correspond-ing respectively to Σ -structure homomorphisms f x : c → Hom E ( E, M ) and f x ′ : d → Hom E ( E, M ) , we have that κ ( c,z ) ◦ x = κ ( d,w ) ◦ x ′ if and only if f x A ( z ) = f x ′ A ( w ) .Now, condition (1) can be formulated by saying that for any generalizedelement x : E → M A (where E ∈ S ) there exists an epimorphic family { e i : E i → E | i ∈ I, E i ∈ S } , a family { ( c i , z i ) | i ∈ I } of objects of thecategory A { x A . ⊤} and generalized elements { y i : E i → H M ( c i ) | i ∈ I } suchthat x ◦ e i = ξ ( c i ,z i ) ◦ y i for every i ∈ I .Under the identification of Proposition 4.5, condition (1) thus rewrites asfollows: for any generalized element x : E → M A (where E ∈ S ) there existsan epimorphic family { e i : E i → E | i ∈ I, E i ∈ S } , a family { ( c i , z i ) | i ∈ I } of objects of the category A { x A . ⊤} and Σ -structure homomorphisms f i : c i → Hom E ( E i , M ) such that f i A ( z i ) = x ◦ e i .Concerning condition (2) , we observe that for any objects ( c, z ) and ( d, w ) of the category A { x A . ⊤} and any Σ -structure homomorphisms f x : c → Hom E ( E, M ) and f x ′ : d → Hom E ( E, M ) (where E ∈ S ) correspondingrespectively to generalized elements x : E → H M ( c ) and x ′ : E → H M ( d ) via the identification of Proposition 4.5, we have that ( x, x ′ ) ∈ R { x A . ⊤} ifand only if there exists an epimorphic family { e i : E i → E | i ∈ I, E i ∈ S } and for each index i ∈ I a T -model a i in K , a Σ -structure homomorphism h i : b i → Hom E ( E i , M ) and two T -model homomorphisms f i : c → b i and f ′ i : d → b i in K such that f i A ( z ) = f ′ i A ( w ) and h i ◦ f i = Hom E ( e i , M ) ◦ f x and h i ◦ f ′ i = Hom E ( e i , M ) ◦ f x ′ .Summarizing, we have the following result. Theorem 5.7.
Let T be a geometric theory, K a small full subcategory of thecategory of set-based models of T , E a Grothendieck topos with a separatingset S and M a T -model in E . Then the following conditions are equivalent:(i) The extension ˜ H M : C T → E of the functor H M : K op → E to thesyntactic category C T (in the sense of section 3.3) satisfies the propertythat the canonical morphism ˜ H M ( M T ) → M is an isomorphism;(ii) For any sort A over Σ , the following conditions are satisfied:(a) For any generalized element x : E → M A (where E ∈ S ) thereexists an epimorphic family { e i : E i → E | i ∈ I, E i ∈ S } andfor each index i ∈ I a T -model c i in K , an element z i of c i A anda Σ -structure homomorphism f i : c i → Hom E ( E i , M ) such that ( f i A )( z i ) = x ◦ e i ;(b) For any two pairs ( c, z ) and ( d, w ) consisting of T -models c and d in K and elements z ∈ cA, w ∈ dA , and any Σ -structure homomor-phisms f : c → Hom E ( E, M ) and f ′ : d → Hom E ( E, M ) (where E is an object of S ), we have that f A ( z ) = f ′ A ( w ) if and only if there xists an epimorphic family { e j : E j → E | j ∈ J, E j ∈ S } and foreach index j ∈ J a T -model b j in K , a Σ -structure homomor-phism h j : b j → Hom E ( E j , M ) and two T -model homomorphisms f j : c → b j and f ′ j : d → b j in K such that f j A ( z ) = f j A ′ ( w ) , h j ◦ f j = Hom E ( e j , M ) ◦ f and h j ◦ f ′ j = Hom E ( e j , M ) ◦ f ′ . Remarks 5.8. (a) If all the T -model homomorphisms in any Grothendiecktopos are monic and T satisfies condition ( ii )( b ) of Theorem 5.3 thencondition ( ii )( b ) of Theorem 5.7 is automatically satisfied.(b) A sufficient condition for condition ( ii )( a ) to hold is that the disjunction ( ⊤ ⊢ x W φ ( x ) ∈I x K φ ( x )) be provable in T , where I x K is the set of geometricformulae in one variable which strongly finitely present a T -model in K (in the sense of Definition 4.7).(c) If for every sort A over Σ the formula { x A . ⊤} strongly presents a T -model F A in K then condition ( i ) of the theorem is automatically sat-isfied; indeed, under this hypothesis for any sort A over Σ the canonicalarrow ˜ H M ( M T ) A = Hom E T -mod ( E ) ( γ ∗E ( F A ) , M ) → M A is an isomorphism(cf. section 4.3).The following result provides an explicit formulation of condition ( ii ) of Theorem 5.1 holding for theories T with respect to small categories K such that every model of K is both strongly finitely presentable and finitelygenerated (with respect to the same generators). Theorem 5.9.
Let T be a geometric theory over a signature Σ and K asmall full subcategory of the category of set-based T -models such that everymodel in K is both strongly finitely presentable and finitely generated (withrespect to the same generators). Then T satisfies condition ( ii ) of Theorem5.1 with respect to the category K if and only if for every model M of T in aGrothendieck topos, the following conditions are satisfied (where P denotesthe set of formulae over Σ which present a model in K ):(i) For any sort A over Σ , the sequent ( ⊤ ⊢ x A _ χ ( ~z ) ∈P ,t A ( ~z ) ( ∃ ~z )( χ ( ~z ) ∧ x = t ( ~z ))) , where the the disjunction is taken over all the formulae χ ( ~z ) in P andall the terms t A ( ~z ) whose output sort is A ;(ii) For any sort A over Σ , any formulae φ ( ~x ) and ψ ( ~y ) in P , where ~x = ( x A , . . . , x A n n ) and ~y = ( y B , . . . , y B m m ) , and any terms t A ( ~x ) and s A ( ~y ) , the sequent φ ( ~x ) ∧ ψ ( ~y ) ∧ t ( ~x ) = s ( ~y ) ⊢ ~x,~y _ χ ( ~z ) ∈P ,p A ( ~z ) ,...,p Ann ( ~z ) q B ( ~z ) ,...,q Bmm ( ~z ) ( ∃ ~z )( χ ( ~z ) ∧∧ ^ i ∈{ ,...,n } ,j ∈{ ,...,m } ( x i = p i ( ~z ) ∧ y j = q j ( ~z ))) , where the disjunction is taken over all the formulae χ ( ~z ) in P and allthe sequences of terms p A ( ~z ) , . . . , p A n n and q B ( ~z ) , . . . , q B m m ( ~z ) whoseoutput sorts are respectively A , . . . , A n , B , . . . , B m and such that, de-noting by ~ξ the set of generators of the model M { ~z.χ } (strongly) finitelypresented by the formula χ ( ~z ) , ( p A ( ~ξ ) , . . . , p A n n ( ~ξ )) ∈ [[ ~x . φ ]] M { ~z.χ } and ( q B ( ~ξ ) , . . . , q B m m ( ~ξ )) ∈ [[ ~y . ψ ]] M { ~z.χ } and t ( p ( ~ξ ) , . . . , p n ( ~ξ )) = s ( q ( ~ξ ) , . . . , q m ( ~ξ )) in M { ~z.χ } . Proof
The proof is analogous to that of Theorem 5.3 and left to the reader. (cid:3)
Remark 5.10.
All the sequents in the statement of the theorem are satisfiedby every model in K . Therefore, adding them to any theory T satisfying thehypotheses of the theorem yields a quotient of T satisfying condition ( ii ) ofTheorem 5.1 with respect to the category K . ( iii ) In this section, we shall give concrete reformulations of conditions ( iii )( a ) - ( b ) - ( c ) of Theorem 5.1.Let us begin our analysis with a proposition which shows that, undersome natural assumptions which are often verified in practice, condition ( iii )( a ) of Theorem 5.1 is satisfied. Proposition 5.11.
Let T be a geometric theory over a signature Σ and let K be a small category of T -mod ( Set ) . Then(i) If T is a quotient of a theory S satisfying condition ( iii )( a ) of Theo-rem 5.1 with respect to a category H of set-based S -models and K is asubcategory of H then T satisfies property ( iii )( a ) of Theorem 5.1 withrespect to K ;(ii) If every T -model in K is strongly finitely presented (in the sense of sec-tion 4.3) then T satisfies condition ( iii )( a ) of Theorem 5.1 with respectto the category K ; iii) If for every T -model c in K the object Hom E T -mod ( E ) ( γ ∗E ( c ) , M ) can bebuilt from c and M by only using geometric constructions (i.e. con-structions only involving finite limits and arbitrary small colimits) then T satisfies condition ( iii )( a ) of Theorem 5.1 with respect to the category K . Proof If T is a quotient of S then for every Grothendieck topos E , thecategory T -mod ( E ) is a full subcategory of the category S -mod ( E ) . Thisclearly implies that for any T -models M and N in a Grothendieck topos E , Hom E T -mod ( E ) ( M, N ) ∼ = Hom E S -mod ( E ) ( M, N ) ; in particular, for any T -model c in K and any T -model M in a Grothendieck topos E , Hom E T -mod ( E ) ( γ ∗E ( c ) , M ) ∼ = Hom E S -mod ( E ) ( γ ∗E ( c ) , M ) . The fact that S satisfies condition ( iii )( a ) of Theo-rem 5.1 with respect to the category H thus implies that T does with respectto the category K , as required.If every T -model c in K is strongly finitely presented (in the sense ofsection 4.3) by a formula φ ( ~x ) over the signature of T then for any T -model M in a Grothendieck topos, Hom E T -mod ( E ) ( γ ∗E ( c ) , M ) ∼ = [[ ~x . φ ]] M (cf. section4.3). Therefore, as the interpretation of geometric formulae is always pre-served by inverse image functors of geometric morphisms, condition ( iii )( a ) of Theorem 5.1 is satisfied by the theory T with respect to the category K .The fact that condition ( iii ) implies condition ( iii )( a ) of Theorem 5.1follows immediately from the fact that geometric constructions are preservedby inverse image functors of geometric morphisms. (cid:3) We shall now proceed to giving concrete reformulations of the conditions ( iii )( b ) - (1) and ( iii )( b ) - (2) of Theorem 5.1 introduced in Remark 5.2(e), inorder to make them more easily verifiable in practice.First, let us explicitly describe, for any object c of K , the arrow η F ( c ) : F ( c ) → Hom E T -mod ( E ) ( γ ∗E ( c ) , ˜ F ( M T )) of condition ( iii )( b ) of Theorem 5.1 in terms of generalized elements.For any generalized element x : E → F ( c ) , η F ( c )( x ) corresponds underthe identification of Proposition 4.5 to the Σ -structure homomorphism z x : c → Hom E ( E, ˜ F ( M T )) defined at each sort A over Σ as the function cA → Hom E ( E, ˜ F ( { x A . ⊤} )) sending any element y ∈ cA to the generalizedelement E → ˜ F ( { x A . ⊤} ) obtained by composing the canonical colimitarrow κ F ( c,y ) : F ( c ) → ˜ F ( { x A . ⊤} ) with the generalized element x : E → F ( c ) .It follows that η F ( c )( x ) is a monomorphism if and only if for any gen-eralized elements x, x ′ : E → F ( c ) , κ F ( c,y ) ◦ x = κ F ( c,y ) ◦ x ′ for every sort A over Σ and element y ∈ cA implies x = x ′ . By Proposition 3.8, thecondition ‘ κ F ( c,y ) ◦ x = κ F ( c,y ) ◦ x ′ ’ is satisfied if and only if there exists an85pimorphic family { e i : E i → E | i ∈ I } in E and for each index i ∈ I a T -model a i in K , a generalized element h i : E i → F ( a i ) and two T -model homomorphisms f i , f ′ i : c → a i in K such that f i A ( y ) = f ′ i A ( y ) and h F ( f i ) , F ( f ′ i ) i ◦ h i = h x, x ′ i ◦ e i .To obtain an explicit characterization of the condition for η F ( c )( x ) to bean epimorphism, we notice that an arrow f : A → B in a Grothendieck topos E is an epimorphism if and only if for every generalized element x : E → B of B there exists an epimorphic family { e i : E i → E | i ∈ I } in E and foreach index i ∈ I a generalized element y i : E i → A such that f ◦ y i = x ◦ e i .Applying this characterization to the arrow η F ( c )( x ) modulo the identifi-cation of Proposition 4.5, we obtain the following criterion: η F ( c )( x ) is anepimorphism if and only if for every object E of E and any Σ -structure ho-momorphism v : c → Hom E ( E, ˜ F ( M T )) , there exists an epimorphic family { e i : E i → E | i ∈ I } in E and for each index i ∈ I a generalized ele-ment x i : E i → F ( c ) such that z x i = Hom E ( e i , ˜ F ( M T )) ◦ v for all i . Thecondition ‘ z x i = Hom E ( e i , ˜ F ( M T )) ◦ v ’ can be explicitly reformulated asthe requirement that for every sort A over Σ and every element y ∈ cA , κ F ( c,y ) ◦ x i = vA ( y ) ◦ e i .Summarizing, we have the following Theorem 5.12.
Let T be a geometric theory over a signature Σ , K be asmall subcategory of T -mod ( Set ) , E a Grothendieck topos with a separatingset S and F : K op → E a flat functor. Then(i) F satisfies condition ( iii )( b ) - (1) of Theorem 5.1 if and only if for any T -model c in K and any generalized elements x, x ′ : E → F ( c ) (where E ∈ S ), if for every sort A over Σ and any element y ∈ cA there existsan epimorphic family { e i : E i → E | i ∈ I, E i ∈ S } and for each index i ∈ I a T -model a i in K , a generalized element h i : E i → F ( a i ) and two T -model homomorphisms f i , f ′ i : c → a i in K such that f i A ( y ) = f ′ i A ( y ) and h F ( f i ) , F ( f ′ i ) i ◦ h i = h x, x ′ i ◦ e i then x = x ′ .(ii) F satisfies condition ( iii )( b ) - (2) of Theorem 5.1 if and only if for any T -model c in K , any object E of S and any Σ -structure homomorphism v : c → Hom E ( E, ˜ F ( M T )) , there exists an epimorphic family { e i : E i → E | i ∈ I, E i ∈ S } and for each index i ∈ I a generalized element x i : E i → F ( c ) such that for every sort A over Σ and any element y ∈ cA , κ F ( c,y ) ◦ x i = vA ( y ) ◦ e i . (cid:3) Under the hypothesis that for any sort A over Σ the formula { x A . ⊤} presents a T -model F A , the model ˜ F ( M T ) is isomorphic to the model inter-preting each sort A with the object F ( F A ) , and the homomorphism z x : c → Hom E ( E, ˜ F ( M T )) corresponding to a generalized element x : E → F ( c ) canbe described as follows: for any sort A over Σ , z x A : cA → Hom E ( E, F ( P A )) y ∈ c A , corresponding to a T -model homomorphism s y : F A → c via the universal property of F A , the generalized element F ( s y ) ◦ x . Therefore, given two generalized elements x, x ′ : E → F ( c ) andan element y ∈ cA , z x A ( y ) = z x ′ A ( y ) if and only if F ( s y ) ◦ x = F ( s y ) ◦ x ′ .Summarizing, we have the following Theorem 5.13.
Let T be a geometric theory over a signature Σ , such that forany sort A over Σ the formula { x A . ⊤} presents a T -model F A , K be a smallsubcategory of T -mod ( Set ) containing the models F A and F : K op → E a flatfunctor with values in a Grothendieck topos E . Then F satisfies condition ( iii )( b ) - (1) of Theorem 5.1 with respect to the category K if and only if forevery T -model c in K the family of arrows { F ( s y ) : F ( c ) → F ( F A ) | A ∈ Σ , , y ∈ cA } is jointly monic. (cid:3) The representation of ˜ F ( M T ) as a filtered E -indexed colimit establishedin section 3.3 allows us to obtain a different reformulation, in terms of the Σ -structure homomorphisms ξ ( c,x ) defined in that context, of Theorem 5.12: Theorem 5.14.
Let T be a geometric theory over a signature Σ , K a smallfull subcategory of T -mod ( Set ) and F : K op → E be a flat functor with valuesin a Grothendieck topos E . Then(i) F satisfies condition ( iii )( b ) - (1) of Theorem 5.1 if and only if for any T -model c in K and any generalized elements x, x ′ : E → F ( c ) , the Σ -structure homomorphisms ξ ( c,x ) and ξ ( c,x ′ ) are equal if and only if x = x ′ .(ii) F satisfies condition ( iii )( b ) - (2) of Theorem 5.1 if and only if for any T -model c in K , any object E of E and any Σ -structure homomorphism z : c → Hom E ( E, ˜ F ( M T )) there exists an epimorphic family { e i : E i → E | i ∈ I } in E and for each index i ∈ I a generalized element x i : E i → F ( c ) such that Hom E ( e i , ˜ F ( M T )) ◦ z = ξ ( c,x i ) for all i ∈ I . Proof ( i ) The equality ξ ( c,x ) = ξ ( c,x ′ ) holds if and only if for every sort A over Σ and any y ∈ cA , ξ ( c,x ) A ( y ) = ξ ( c,x ′ ) A ( y ) , i.e. κ ( c,y ) ◦ x = κ ( c,y ) ◦ x ′ .Now, by Proposition 3.8, this latter condition is satisfied if and only if thereexists an epimorphic family { e i : E i → E | i ∈ I } in E and for each index i ∈ I a T -model a i in K , a generalized element h i : E i → F ( a i ) and two T -model homomorphisms f i , f ′ i : c → a i such that f i A ( y ) = f ′ i A ( y ) and h F ( f i ) , F ( f ′ i ) i ◦ h i = h x, x ′ i ◦ e i , as required. ( ii ) It suffices to notice that the condition that for every sort A over Σ and every element y ∈ cA , κ F ( c,y ) ◦ x i = zA ( y ) ◦ e i can be reformulated as therequirement that Hom E ( e i , ˜ F ( M T )) ◦ z = ξ ( c,x i ) (for any i ∈ I ). (cid:3) ( iii )( b ) - (1) of Theorem5.1, we can prove that if all the arrows in C are sortwise monic then T alwayssatisfies the condition.First, we need a lemma. Lemma 5.15.
Under the hypotheses specified above, if all the homomor-phisms in the category K are sortwise monic then for any pair ( c, x ) consist-ing of an object c of K and of a generalized element x : E → F ( c ) such that E ≇ E , the Σ -structure homomorphism ξ ( c,x ) : a → Hom E ( E, ˜ F ( M T )) issortwise injective. Proof
For any sort A over Σ , the function ξ ( c,x ) A : cA → Hom E ( E, M A ) sends any element y ∈ cA to the generalized element κ ( c,y ) ◦ x . Now, for any y , y ∈ cA , we have, by Proposition 3.8, that κ ( c,y ) ◦ x = κ ( c,y ) ◦ x if andonly if there exists an epimorphic family { e i : E i → E | i ∈ I } in E and foreach index i ∈ I a T -model a i in K , a generalized element h i : E i → F ( a i ) and a T -model homomorphism f i : c → a i in K such that f i A ( y ) = f i A ( y ) and F ( f i ) ◦ h i = x ◦ e i . If E ≇ E then the set I is non-empty, that is thereexists i ∈ I ; thus we have that f i A ( y ) = f i A ( y ) , which entails y = y as f i is sortwise monic by our hypothesis. (cid:3) Corollary 5.16.
Let T be a geometric theory over a signature Σ , K a smallfull subcategory of T -mod ( Set ) whose arrows are all sortwise monic homo-morphisms. Then T satisfies condition ( iii )( b ) - (1) of Theorem 5.1. Proof
By Proposition 3.11, for any x : E → F ( c ) and x ′ : E → F ( c ) ,the following ‘joint embedding property’ holds: there exists an epimorphicfamily { e i : E i → E | i ∈ I } in E and for each index i ∈ I a T -model c i in K , homomorphisms f i : c → c i , g i : c → c i in K and a generalized element x i : E i → F ( c i ) such that h x, x ′ i ◦ e i = h F ( f i ) , F ( g i ) i ◦ x i , Hom E ( E i , M ) ◦ ξ ( c,x ) = ξ ( c i ,x i ) ◦ f i and Hom E ( E i , M ) ◦ ξ ( c,x ′ ) = ξ ( c i ,x i ) ◦ g i (for all i ∈ I ).Clearly, we can suppose without loss of generality all the objects E i to benon-zero.Now, since all the arrows in K are sortwise monic homomorphisms, byLemma 5.15 for each i ∈ I the arrow ξ ( c i ,x i ) is sortwise monic and hence ξ ( c,x ) = ξ ( c,x ′ ) implies f i = g i . Therefore x ◦ e i = x ′ ◦ e i for all i ∈ I (since h x, x ′ i ◦ e i = h F ( f i ) , F ( g i ) i ◦ x i ), and hence x = x ′ , as required. (cid:3) We shall now proceed to identifying some natural sufficient conditions fora theory T to satisfy condition ( iii )( b ) - (2) of Theorem 5.1. Before statingthe relevant theorem, we need a number of preliminary results. Lemma 5.17.
Let Σ be a signature without relation symbols and T a geo-metric theory over a signature Σ ′ obtained from Σ by solely adding relationsymbols whose interpretation in any T -model in a Grothendieck topos is the omplement of the interpretation of a geometric formula over Σ (for instance,the injectivization of a geometric theory over Σ in the sense of Definition6.33). Let f : M → N and g : P → N be homomorphisms of T -models ina Grothendieck topos E , and let k be an assignment to any sort A over Σ ofan arrow kA : M A → P A such that gA ◦ kA = f A . Then, if g is sortwisemonic, k defines a T -model homomorphism M → P such that g ◦ k = f . Proof
The fact that k preserves the interpretation of function symbolsover Σ ′ follows from the fact that f and g do, as g is sortwise monic. Itremains to prove that for any relation symbol R A , . . . , A n over Σ ′ andany generalized element x : E → M A × · · · × M A n in E , if x factorsthrough R M M A × · · · × M A n then ( kA × · · · × kA n ) ◦ x factorsthrough R P P A × · · · × P A n . Now, if R P is the complement of theinterpretation i φ : [[ φ ( ~x )]] P P A × · · · × P A n of a geometric formula φ ( ~x ) = φ ( x A , . . . , x A n ) in the model P , this latter condition is equivalent tothe requirement that the equalizer e of ( kA × · · · × kA n ) ◦ x and i φ be zero;but, since g is a Σ ′ -structure homomorphism, the subobject e is containedin the equalizer of the arrows ( gA × · · · × gA n ) ◦ ( kA × · · · × kA n ) ◦ x =( f A × · · · × f A n ) ◦ x and [[ φ ( ~x )]] N N A × · · · × N A n , which is zero since f is a Σ ′ -structure homomorphism and R N is the complement of [[ φ ( ~x )]] N .Therefore e ∼ = 0 E , as required. (cid:3) Lemma 5.18.
Let E be a Grothendieck topos. Then the inverse image func-tor γ ∗E : Set → E of the unique geometric morphism γ E : E →
Set is faithfulif and only if E is non-trivial (i.e., E ≇ E ). Proof
It is clear that if E is trivial then γ ∗E : Set → E is not faithful, itbeing the constant functor with value E .In the converse direction, suppose that E is non-trivial. Given two func-tions f, g : A → B in Set , the arrows γ ∗E ( f ) , γ ∗E ( g ) : ` a ∈ A E → ` b ∈ B E in E arecharacterized by the following identities: for any a ∈ A γ ∗E ( f ) ◦ s a = t f ( a ) and γ ∗E ( g ) ◦ s a = t g ( a ) , where s a : 1 → ` a ∈ A E and t b : 1 → ` b ∈ B E are re-spectively the a -th and b -th coproduct arrows (for any a ∈ A and b ∈ B );so γ ∗E ( f ) = γ ∗E ( g ) if and only if for every a ∈ A , t f ( a ) = t g ( a ) . Now, since asubobject with domain E in a non-trivial topos E cannot be disjoint fromitself, it follows that f ( a ) = g ( a ) for every a ∈ A . Therefore f = g , asrequired. (cid:3) Corollary 5.19.
Let T be a geometric theory over a signature Σ , E a non-trivial Grothendieck topos (i.e., E ≇ E ), M and N two T -models in Set and f a function sending each sort A over Σ to a map f A : M A → N A in such a way that the assignment A → γ ∗E ( f A ) : γ ∗E ( M A ) → γ ∗E ( N A ) is a T -model homomorphism in E . Then the assignment A → f A is a T -modelhomomorphism M → N in Set . roof We have to prove that f preserves the interpretation of functionsymbols over Σ and the satisfaction of atomic relations over Σ . Since thepreservation of the interpretation of function symbols over Σ can be expressedas the commutativity of a certain square involving finite products of sets ofthe form M A and
N A , its satisfaction follows from that of γ ∗E ( f ) by virtue ofLemma 5.18. It remains to prove that f preserves the satisfaction of atomicrelations over Σ . Let R A , . . . , A n be a relation symbol over Σ . From thefact that γ ∗E ( f ) : γ ∗E ( M ) → γ ∗E ( N ) preserves the satisfaction by R it followsthat for any n -tuple ~a = ( a , . . . , a n ) ∈ R M , the coproduct arrow : 1 E → ` ( b ,...,b n ) ∈ NA ×···× NA n E corresponding to the n -tuple ( f A ( a ) , . . . , f A n ( a n )) factors through the subobject ` ( b ,...,b n ) ∈ R N E ` ( b ,...,b n ) ∈ NA ×···× NA n E . Butthis immediately implies, since distinct coproduct arrows are disjoint fromeach other and the topos E is non-trivial, that f ( ~a ) ∈ R N , as required. (cid:3) Corollary 5.20.
Let Σ be a signature without relation symbols and T ageometric theory over a signature Σ ′ obtained from Σ by solely adding relationsymbols whose interpretation in any T -model in a Grothendieck topos is thecomplement of the interpretation of a geometric formula over Σ (for instance,the injectivization of a geometric theory over Σ in the sense of Definition6.33). Let K be a small full subcategory of T -mod ( Set ) , a and b T -modelsin K , M a T -model in a Grothendieck topos E and f : a → Hom E ( E, M ) , g : b → Hom E ( E, M ) Σ ′ -structure homomorphisms, where E is an objectof E . Let k be an assignment sending any sort A over Σ to a function kA : aA → bA such that gA ◦ kA = f A . Suppose that either(i) there are no relation symbols in Σ ′ except possibly for a binary relationsymbol which is T -provably complemented to the equality relation on asort of Σ and the Σ ′ -structure homomorphisms f and g are sortwisemonic; or(ii) E ≇ E and the T -model homomorphism ˜ g : γ ∗E /E ( b ) → M correspond-ing to g under the identification of Proposition 4.5 is sortwise monic.Then the assignment A → kA defines a T -model homomorphism a → b in Set . Proof
We have to verify that k preserves the interpretation of functionsymbols over Σ ′ as well as the satisfaction by any binary relation symbolwhich is T -provably complemented to the equality relation on some sort over Σ . By Lemma 6.34 below, k preserves the interpretation of any binary re-lation symbol which is T -provably complemented to the equality relation onsome sort over Σ if and only if it is sortwise monic. This holds under hypoth-esis ( i ) since f is sortwise monic and for every sort A over Σ , gA ◦ kA = f A .90n the other hand, under hypothesis ( i ) , the fact that k preserves the inter-pretation of function symbols over Σ ′ follows from the fact that g and f do,since g is sortwise monic and gA ◦ kA = f A for every sort A over Σ . Thisshows that our thesis is satisfied under hypothesis ( i ) .Let us now suppose that hypothesis ( ii ) holds. Consider the T -modelhomomorphisms ˜ f : γ ∗E /E ( a ) → M and ˜ g : γ ∗E /E ( b ) → M corresponding to f and g under the identification of Proposition 4.5. Clearly, the assignment A → ˜ kA = γ ∗E /E ( kA ) satisfies ˜ gA ◦ ˜ kA = ˜ f A for all sorts A over Σ . Itthus follows from Lemma 5.17 that A → ˜ kA is a T -model homomorphism γ ∗E /E ( a ) → γ ∗E /E ( b ) ; but this in turn implies, by Corollary 5.19 (notice that E ≇ E if and only if the topos E /E is non-trivial), that the assignment A → kA is a T -model homomorphism a → b , as required. (cid:3) Theorem 5.21.
Let Σ be a signature without relation symbols and T a ge-ometric theory over a signature Σ ′ obtained from Σ by solely adding rela-tion symbols whose interpretation in any T -model in a Grothendieck toposis the complement of the interpretation of a geometric formula over Σ (forinstance, the injectivization of a geometric theory over Σ in the sense of Def-inition 6.33). Let K be a small full subcategory of T -mod ( Set ) whose objectsare all finitely generated T -models. Suppose that(i) either all the arrows in K are sortwise monic homomorphisms and thereare no relation symbols except possibly for a binary relation symbolwhich is T -provably complemented to the equality relation on a givensort of Σ or(ii) all the T -model homomorphisms in any Grothendieck topos are sortwisemonic.Then T satisfies condition ( iii )( b ) - (2) of Theorem 5.1 with respect to thecategory K . Proof
By Theorem 5.14, we have to verify that for any T -model c in K , anyobject E of E and any Σ -structure homomorphism z : c → Hom E ( E, ˜ F ( M T )) there exists an epimorphic family { e i : E i → E | i ∈ I } in E and for eachindex i ∈ I a generalized element x i : E i → F ( c ) such that Hom E ( e i , M ) ◦ z = ξ ( c,x i ) for all i ∈ I .If E ∼ = 0 E then the condition is trivially satisfied; indeed, one can take I = ∅ . We shall therefore suppose E ∼ = 0 E or, equivalently, that the topos E /E is non-trivial.From now on, we shall suppose for simplicity that Σ is one-sorted, butall our arguments can be straightforwardly extended to the general case.Let { r , . . . , r n } be a set of generators for the model c . Consider theirimages z ( r ) , . . . , z ( r n ) : E → M under the homomorphism z . By Propo-sition 3.11, for any i ∈ { , . . . , n } there exists an epimorphic family { e ij : ij → E | j ∈ I i } in E and for each index j ∈ J i a T -model a ij in K , ageneralized element x ij : E ij → F ( a ij ) and an element y ij ∈ a ij such that ξ ( a ij ,x ij ) ( y ij ) = z ( r i ) ◦ e ij .For any tuple ~k = ( k , . . . , k n ) ∈ J × · · · × J n , consider the iteratedpullback e ~k : E ~k =: E k × E · · · × E E nk n → E . The family of arrows { e ~k : E ~k → E | ~k ∈ J × · · · × J n } is clearly epimorphic. For any i ∈ { , . . . , n } and k i ∈ J i , set x i~k : E ~k → F ( a ik i ) equal to the composite of the generalizedelement x ik i : E ik i → F ( a ik i ) with the canonical pullback arrow p ik i : E ~k → E ik i . For any fixed ~k ∈ J × · · · × J n , by inductively applying the jointembedding property of Proposition 3.11, we can find an epimorphic family { u ~kl : U ~kl → E ~k | l ∈ L ~k } and for each index l ∈ L ~k a T -model d ~kl in K , ageneralized element x ~kl : U ~kl → F ( d ~kl ) , and arrows f ~kl : a ik i → d ~kl in K suchthat x i~k ◦ u ~kl = F ( f ~kl ) ◦ x ~kl (for all i ∈ { , . . . , n } and l ∈ L ~k ).Let us prove that z ( r i ) ◦ e ~k ◦ u ~kl = ξ ( d ~kl ,x ~kl ) ( f ~kl ( y ik i )) (for any i, ~k and l ).We have already observed that for any i ∈ { , . . . , n } , we have z ( r i ) ◦ e ik i = ξ ( a iki ,x iki ) ( y ik i ) . Composing both sides of the equation with p ik i and applyingLemma 3.10(ii) yields the equality z ( r i ) ◦ e ~k = ξ ( a iki ,x i~k ) ( y ik i ) . On the otherhand, by Lemma 3.10(i) we have that ξ ( a iki ,x i~k ) ( y ik i ) ◦ u ~kl = ξ ( d ~kl ,x ~kl ) ( f ~kl ( y ik i )) .Our thesis thus follows by combining the former identity with the one ob-tained by composing both sides of the latter identity with u ~kl .Let us now consider the Σ ′ -structure homomorphisms Hom E ( e ~k ◦ u ~kl , ˜ F ( M T )) ◦ z : c → Hom E ( U ~kl , ˜ F ( M T )) and ξ ( d ~kl ,x ~kl ) : d ~kl → Hom E ( U ~kl , ˜ F ( M T )) . Since, under either assumption ( i ) or ( ii ) , the arrows of the category K are sortwise monic homomorphisms, by Proposition 5.15 the homomorphism ξ ( d ~kl ,x ~kl ) is injective. Therefore, since the image of all the generators r , . . . , r n of c under the homomorphism Hom E ( e ~k ◦ u ~kl , ˜ F ( M T )) ◦ z is contained in theimage of the homomorphism ξ ( d ~kl ,x ~kl ) , an easy induction on the structure t ( r , . . . , r n ) of the elements of c shows that the image of all elements of c belongs to the image of ξ ( d ~kl ,x ~kl ) ; in other words, there exists a factorization k ~kl : c → d ~kl of Hom E ( e ~k ◦ u ~kl , ˜ F ( M T )) ◦ z across ξ ( d ~kl ,x ~kl ) . By Corollary 5.20,such factorization is a T -model homomorphism. We have thus found anepimorphic family on E , namely { e ~k ◦ u ~kl | ~k ∈ J × · · · × J n , l ∈ L ~k } , andfor every ~k and l a T -model homomorphism k ~kl : c → d ~kl in K such that Hom E ( e ~k ◦ u ~kl , ˜ F ( M T )) ◦ z = ξ ( d ~kl ,x ~kl ) ◦ k ~kl . x ~kl ′ : U ~kl → F ( c ) of the generalized elements x ~kl : U ~kl → F ( d ~kl ) with the arrows F ( k ~kl ) thus yield generalized elements such that Hom E ( e ~k ◦ u ~kl , ˜ F ( M T )) ◦ z = ξ ( d ~kl ,x ~kl ) ◦ k ~kl = ξ ( c,x ~kl ′ ) , where the last equal-ity follows by Lemma 3.10(i). This completes our proof. (cid:3) Remark 5.22.
The assumption that all the arrows in K be sortwise monicin condition ( i ) is weaker in general than the requirement of condition ( ii ) that all the T -model homomorphisms in any Grothendieck topos be sortwisemonic. Anyway, condition ( ii ) is a necessary condition for T to be classifiedby the topos [ K , Set ] if all the arrows in K are sortwise monic (cf. Corollary6.9). Corollary 5.23.
Let T be a geometric theory satisfying the hypotheses ofTheorem 5.21 with respect to a small full subcategory K of T -mod ( Set ) . Then T is of presheaf type classified by the topos [ K , Set ] if and only if it satisfiescondition ( i ) of Theorem 5.1 and condition ( ii )( a ) of Theorem 5.7. Proof
Condition ( iii ) of Theorem 5.1 is satisfied by Theorem 5.21, whilethe first part of condition ( ii ) holds by Remark 5.8(a). (cid:3) Thanks to the general theory of indexed filtered colimits developed in section2, we can reformulate the conditions of Theorem 5.1 in more abstract, thoughless explicit, terms as follows.Let T be a geometric theory over a signature Σ and K a small full sub-category of T -mod ( Set ) .Condition ( i ) of Theorem 5.1 for T with respect to K can be reformulatedas the requirement that for every model M of T in a Grothendieck topos E ,the E -indexed category of elements R f H M of the functor H M (or equivalentlyits E -final subcategory R H M E , cf. Theorem 2.13) be E -filtered.For any sort A over Σ , we have a functor P A : K →
Set of evaluationof models in K at the sort A . Using the notation of Theorem 2.5, we havean associated internal diagram ( P A ) E in [ K , E ] and hence E -indexed functor K E → E E . These functors, with A varying among the sorts over Σ , clearlylift to a E -indexed functor P : K E → T -mod ( E ) .In these terms, condition ( ii ) of Theorem 5.1 for the model M can bereformulated as the requirement that the canonical E -indexed cone over the E -indexed functor P ◦ π fH M be colimiting (cf. the proof of Proposition 3.11).Conditions ( i ) and ( ii ) can thus be interpreted by saying that every T -model M in a Grothendieck topos E is canonically a E -filtered colimit ofconstant finitely presentable T -models in Set which belong to K .Under the assumption that conditions ( i ) and ( ii ) are satisfied, condi-tion ( iii ) is equivalent to the requirement that for any T -model c in K and93ny Grothendieck topos E , the T -model γ ∗E ( c ) be E -finitely presentable (inthe sense of section 4.4) or, more weakly, that the internal E -indexed homfunctor Hom E T -mod ( E ) ( γ ∗E ( c ) , − ) : T -mod ( E ) → E E preserve E -filtered colimitsof diagrams of the form γ E ◦ D , where D is a diagram defined on a ( E -finalsubcategory of a) small internal filtered category. Indeed, the fact that thiscondition is necessary for T to be of presheaf type follows from Theorem4.13, while the fact that it is sufficient, together with conditions ( i ) and ( ii ) of Theorem 5.1, for T to be classified by the topos [ K , Set ] can be provedby showing that it implies condition ( iii )( a ) of Theorem 5.1. In fact, thisimmediately follows from the fact that inverse image functors of geometricmorphisms preserve internal (filtered) colimits in view of the fact that, bycondition ( ii ) , every T -model M in a Grothendieck topos E is canonically a E -filtered colimit of ‘constant’ finitely presentable T -models in K .Overall, we can conclude that a geometric theory T is of presheaf type ifand only if every T -model in any Grothendieck topos E is a E -filtered colimitof its canonical diagram made of ‘constant’ finitely presentable models whichare E -finitely presentable. In this section we introduce the notion of faithful interpretation of geometrictheories and establish sufficent criteria for faithful interpretations of theoriesof presheaf type to be again of presheaf type. We shall treat conditions (i),(ii) and (iii) of Theorem 5.1 separately.
Definition 6.1.
A geometric morphism a : Set [ T ] → Set [ T ′ ] between theclassifying toposes of two geometric theories is said to be a faithful interpre-tation of T ′ into T if the induced morphism a E : T -mod ( E ) → T ′ -mod ( E ) of categories of models is faithful, reflects isomorphisms and the equalityrelation between objects.For any faithful interpretation of T ′ into T and any Grothendieck topos E , we have a subcategory Im ( a E ) of T ′ -mod ( E ) whose objects (resp. arrows)are exactly the models (resp. the model homomorphisms) in the image ofthe functor a E . For any functor F : A → T -mod ( E ) , F is full and faithfulif and only if its composite a E ◦ F with a E is full and faithful as a functor A → Im ( a E ) . 94 emarks 6.2. (a) Any quotient S of a geometric theory T defines a faithfulinterpretation of T into S ;(b) Any expansion (in the sense of section 7.3) T of a geometric theory T ′ over a signature which does not contain new sorts with respect to thesignature of T defines a faithful interpretation of T ′ into T ; in particular,the injectivization T m (in the sense of Definition 6.33) of a geometrictheory T defines a faithful interpretation of T into T m .Given a faithful interpretation of T ′ into T as above, we can reformulatethe conditions of Theorem 5.1 for the theory T with respect to a small fullsubcategory K of T -mod ( Set ) in alternative ways, as follows.Condition ( iii )( c ) of Theorem 5.1 for T with respect to K can be formu-lated as follows: the composite functor q E = a E ◦ u K : Flat ( K op , E ) → T ′ -mod ( E ) is full and faithful into the image Im ( a E ) of a E , where u K is the functor Flat ( K op , E ) → T -mod ( E ) induced by the canonical geometric morphism p K : [ K , Set ] → Set [ T ] .Condition ( ii ) of Theorem 5.1 for T with respect to K can be reformulatedby saying that for any T -model M in a Grothendieck topos E , the canonicalmorphism q E ( H M ) → a E ( M ) is an isomorphism.Suppose moreover that the functor a Set : T -mod ( Set ) → T ′ -mod ( Set ) restricts to a functor f : K → H , where H is a small full subcategory of T ′ -mod ( Set ) .We have a commutative diagram [ K , Set ] p K (cid:15) (cid:15) [ f, Set ] / / [ H , Set ] p H (cid:15) (cid:15) Set [ T ] a / / Set [ T ′ ] , where p K (resp. p H ) is the canonical geometric morphism determined by theuniversal property of the classifying topos of T (resp. of T ′ ).Also, for any T ′ -model M in a Grothendieck topos E , the functors a E induce a E -indexed functor Z a : Z H T M → Z H T ′ a E ( M ) E -indexed categories of elements R H T M and R H T ′ a E ( M ) of thefunctors H T M := Hom E T -mod ( E ) ( γ ∗E ( − ) , M ) : K op → E and H T ′ a E ( M ) := Hom E T ′ -mod ( E ) ( γ ∗E ( − ) , a E ( M )) : H op → E . Since a is by our hypothesis a faithful interpretation of theories, R a isfaithful and reflects equalities and isomorphisms. Suppose now that R a ismoreover E -full (in the sense of Definition 2.9). Since for any functor F withvalues in a Grothendieck topos, F is flat if and only if its E -indexed categoryof elements is E -filtered, we conclude by Proposition 2.11 that if R a is E -finalthen the theory T satisfies condition (i) of Theorem 5.1 with respect to thecategory K if T ′ does with respect to the category H . Moreover, Proposition3.14 and Theorem 2.12 ensure that if T ′ satisfies condition (ii) of Theorem5.1 with respect to the category H then T does with respect to the category K . Summarizing, we have the following result. Theorem 6.3.
Let a : Set [ T ] → Set [ T ′ ] be a faithful interpretation ofgeometric theories and let K and H be small subcategories respectively of T -mod ( Set ) and of T ′ -mod ( Set ) such that the functor a Set : T -mod ( Set ) → T ′ -mod ( Set ) restricts to a functor K → H and the E -indexed functor Z a : Z H T M → Z H T ′ a E ( M ) is E -full and satisfies the equivalent conditions of Proposition 6.5.Then the theory T satisfies condition ( i ) (resp. condition ( ii ) ) of Theorem5.1 with respect to the category K if T ′ satisfies condition ( i ) (resp. condition ( ii ) ) of Theorem 5.1 with respect to the category H . (cid:3) Remark 6.4. If T is the injectivization of T ′ (in the sense of Definition 6.33)or if T is a quotient of T ′ then the functor R a is E -full. Indeed, in the firstcase this follows from the fact that if the composite of two arrows is monicthen the first one is monic, while in the second it follows from the fact thatthe category of models of T is a full subcategory of the category of modelsof T ′ .The following proposition provides an explicit rephrasing of the finalitycondition for the functor R a . 96 roposition 6.5. Let a : Set [ T ] → Set [ T ′ ] be a faithful interpretation ofgeometric theories let M be a T -model in a Grothendieck topos E such thatthe functor H T ′ a E ( M ) is flat. Then the following conditions are equivalent:(a) the functor R a is E -final;(b) For any object E of E , any T ′ -model c in H , any T -model M in E andany T ′ -model homomorphism x : γ ∗E /E ( c ) → ! ∗ E ( a E ( M )) , there exists anepimorphic family { e i : E i → E | i ∈ I } in E and for each i ∈ I a T ′ -model homomorphism f i : c → a Set ( c i ) , where c i is a T -model in K ,and a T ′ -model homomorphism x i : γ ∗E /E i ( a Set ( c i )) → ! E i ( a E ( M )) suchthat x i ◦ γ ∗E /E i ( f i ) = e ∗ i ( x ) for all i ∈ I ;(c) For any object E of E and any Σ -structure homomorphism x : c → Hom E ( E, a E ( M )) , where c is a T ′ -model in H , there exists an epimor-phic family { e i : E i → E | i ∈ I } in E and for each i ∈ I a T ′ -modelhomomorphism f i : c → a Set ( c i ) , where c i is a T -model in K , and a Σ -structure homomorphism x i : a Set ( c i ) → Hom E ( E i , a E ( M )) such that x i ◦ f i = Hom E ( e i , a E ( M )) ◦ x for all i ∈ I . Proof
The equivalence between the first two conditions immediately followsfrom the fact that, H M being flat, the E -indexed category R H M is E -filteredand hence Remark 2.10(c) applies, while the equivalence between the secondand the third condition follows from Proposition 4.5. (cid:3) Remark 6.6. If E = Set and T is a quotient of T ′ the condition rewrites asfollows: for any T -model M in Set , any model c in H and any T ′ -model ho-momorphism f : c → M , there exists a T -model d in K and homomorphisms g : c → d and h : d → M such that h ◦ g = f .In the particular case where T ′ is the empty theory over a signature Σ , T is a F -finitary geometric theory over Σ (in the sense of [32]), H is thecategory of finite Σ -structures and K is the category of finite T -models, thecondition, required for every model M of T in Set , specializes precisely tothe ‘finite structure condition’ of [32].The following result shows a relation between the action on points of amorphism of classifying toposes and a related induced action on flat functors.
Theorem 6.7.
Let a : Set [ T ] → Set [ T ′ ] be an interpretation of geometrictheories and K and H small subcategories respectively of T -mod ( Set ) and of T ′ -mod ( Set ) such that the functor a Set : T -mod ( Set ) → T ′ -mod ( Set ) restricts to a functor f : K → H . Then for any Grothendieck topos E theextension functor Flat ( H op , E ) → Flat ( K op , E ) → T -mod ( E ) long the geometric morphism [ f, Set ] : [ K , Set ] → [ H , Set ] takes values inthe full subcategory Im ( a E ) of T -mod ( E ) . Proof
The diagram [ H , Set ] p H / / Set [ T ′ ][ K , Set ] [ f, Set ] O O p K / / Set [ T ] a O O clearly commutes by definition of the functor f . Therefore, in view of Di-aconescu’s theorem, for any Grothendieck topos E we have a commutativediagram Flat ( H op , E ) u T ′ ( H , E ) / / T ′ -mod ( E ) Flat ( K op , E ) ξ E O O u T ( K , E ) / / T -mod ( E ) , a E O O where u T ( K , E ) and u T ′ ( H , E ) are the functors of section 3.3 and ξ E is the extensionfunctor induced by the geometric morphism [ f, Set ] , from which our thesisimmediately follows. (cid:3) The following corollary can be obtained as a particular case of the the-orem by taking T ′ to be the geometric theory of flat functors on C op , T tobe its injectivization (in the sense of Definition 6.33), H equal to C and K equal to D . Corollary 6.8.
Let D be a small subcategory of a small category C whosearrows are all monic (in C ), and E a Grothendieck topos. Then(a) For any object d ∈ D , ˜ F ( d ) is a decidable object of E ;(b) For any natural transformation α : F → G between two flat functors F, G : D op → E , ˜ α ( c ) : ˜ F ( c ) → ˜ G ( c ) is monic in E for every c ∈ C . Inparticular, α ( d ) : F ( d ) → G ( d ) is monic in E for every d ∈ D . (cid:3) The following corollary is obtained by applying the theorem in the case T is equal to the injectivization of T ′ (in the sense of Definition 6.33) and H = K = f.p. T ′ -mod ( Set ) . Corollary 6.9.
Let T be a theory of presheaf type such that all the T -modelhomomorphisms between its finitely presentable (set-based) models are sort-wise injective. Then every T -model homomorphism in every Grothendiecktopos is sortwise monic. (cid:3) .2 Finitely presentable and finitely generated models In this section, we discuss, for the purpose of establishing our main criteria forthe injectivization of a theory of presheaf type to be again of presheaf type,the relationship between finitely presentable and finitely generated modelsof a given geometric theory.Throughout this section, we shall assume for simplicity all our first-ordersignatures to be one-sorted, if not otherwise stated. It is certainly possibleto extend our definitions and results to the general situation, but we shallnot embark in the straightforward task of making this explicit.Let Σ be a first-order signature. Recall from [20] that, given a Σ -structure M and a subset A ⊆ M , the Σ -structure h A i generated by A is the small-est Σ -substructure of M containing A ; by the proof of Theorem 1.2.3 [20], h A i can be concretely represented as the union S n ∈ N A n , where the subsets A n are defined inductively as follows: A = A ∪ { c M } , where c M are theinterpretations in M of the constants over Σ , and A n +1 = A n ∪ { f M ( ~a ) : for some n > , f is a n -ary function symbol of Σ and ~a is a n -tuple of ele-ments of A n }. A Σ -structure M is said to be finitely generated if there existsa finite subset A ⊆ M such that M = h A i .In the category T -mod ( Set ) of set-based models of a geometric theory T , we say that an object B is a quotient of an object A if there exists asurjective T -model homomorphism A → B . Proposition 6.10.
Let T be a geometric theory over a signature Σ . Thenany quotient in T -mod ( Set ) of a finitely generated T -model is finitely gener-ated. Proof
Let q : M → N be a surjective homomorphism of T -models. Supposethat M is finitely generated, by a finite subset A ⊆ M . Set B = q ( A ) . Thisset is clearly finite, and N is equal to h B i . To prove this we show, by induc-tion on n , that for any n ∈ N , q ( A n ) = B n . As q is a Σ -structure homomor-phism, we have that q ( A ) = B and if q ( A n ) = B n then clearly q ( A n +1 ) = B n +1 . Now, as q is surjective, N = q ( A ) = q ( S n ∈ N A n ) = S n ∈ N q ( A n ) = S n ∈ N B n ,as required. (cid:3) Proposition 6.11.
Let T be a geometric theory over a signature Σ whoseaxioms are all of the form ( φ ⊢ ~x ψ ) , where ψ is a quantifier-free geometricformula. Then any finitely presentable T -model is finitely generated. Proof
First, we notice that any geometric theory over a signature Σ suchthat all its axioms are of the form ( φ ⊢ ~x ψ ) , where ψ is a quantifier-freegeometric formula, satisfies the property that any substructure of a modelof T is a model of T . 99et M be a finitely presentable T -model. We can clearly represent M asa filtered colimit (actually, directed union) of its finitely generated substruc-tures (equivalently, submodels, cf. the above remark). By definition of finitepresentability, the identity on M factors through one of the embeddings of afinitely generated T -submodel of M into M ; such embedding is thus neces-sarily an isomorphism, and M coincides with its domain. In particular, M is finitely generated, as required. (cid:3) Proposition 6.12.
Let T be a geometric theory over a signature Σ whosecategory of set-based models is finitely accessible (in particular, a theory ofpresheaf type). Then any finitely generated T -model is a quotient of a finitelypresentable models of T in the category T -mod ( Set ) .If moreover all the axioms of T are of the form ( φ ⊢ ~x ψ ) , where ψ isa quantifier-free geometric formula, then the finitely generated T -models areprecisely the quotients of the finitely presentable models of T in the category T -mod ( Set ) . Proof
As the category T -mod ( Set ) is finitely accessible, M can be expressedas a filtered colimit ( M = colim ( N i ) , { J i : N i → M | i ∈ I} ) of finitelypresentable models N i . Let a , . . . , a n a finite set of generators for M . Asthe family of arrows { J i : N i → M | i ∈ I} is jointly surjective (sincefiltered colimits in T -mod ( Set ) are computed pointwise as in Set ), for anygenerator a j there exists an index k j ∈ I and an element d j of N k j suchthat J k j ( d j ) = a j . Now, the set of generators a j being finite and the indexcategory I being filtered, we can suppose without loss of generality that allthe k j are equal. We thus have an index k ∈ I and a string of elements of N k which are sent by J k to the generators of M ; hence the arrow J k : N k → M is surjective.The second part of the proposition follows from Propositions 6.11 and6.10, which ensure that every quotient of a finitely presentable model isfinitely generated. (cid:3) Proposition 6.13.
Let Σ be a signature without relation symbols and T ageometric theory over a larger signature Σ ′ obtained from Σ by solely addingrelation symbols whose interpretation in any set-based T -model coincides withthe complement of a geometric formula over Σ (for instance, the injectiviza-tion of a geometric theory over Σ ). Suppose that all the T -model homomor-phisms between set-based T -models are injective. Then any finitely generated T -model is finitely presentable. Proof
From Proposition 6.11, we know that every finitely presentable T -model is finitely generated. It thus remains to prove the converse.We observed in [6] (Lemma 6.2) that, given a category D with filteredcolimits, an object M of D is finitely presentable in D if and only if for any100ltered diagram D : I → D , any arrow M → colim ( D ) factors throughone of the canonical colimit arrows J i : D ( i ) → colim ( D ) and for any ar-rows f : M → D ( i ) and g : M → D ( j ) in D such that J i ◦ f = J j ◦ g ,there exists k ∈ I and two arrows s : i → k and t : j → k such that D ( s ) ◦ f = D ( t ) ◦ g . Now, the latter condition is automatically satisfied ifall the arrows of D are monic (since the category I is filtered); in particu-lar, applying this criterion to the case of our theory T yields the followingcharacterization of the finitely presentable T -models: a set-based T -model M is finitely presentable if and only if for any filtered colimit colim ( N i ) ofset-based T -models, with colimit arrows J i : N i → colim ( N i ) , any T -modelhomomorphism f : M → colim ( N i ) factors through at least one arrow J i inthe category T -mod ( Set ) .Now, if M is finitely generated by elements a , . . . , a n , for any of gener-ator a i of M there exists an index k i of I and an element b i ∈ N k i such that f ( a i ) = J i ( b i ) . As the category I is filtered, we can suppose without loss ofgenerality that all the k i are equal to each other. Therefore there exists anindex k ∈ I and elements b , . . . , b n of N k such that f ( a i ) = J k ( b i ) for all i .The image of M under f is thus entirely contained in N k and hence we havea function g : M → N k such that J k ◦ g = f . Let us prove that g is a Σ ′ -structure homomorphism. The fact that g is a Σ -structure homomorphism,i.e. that it commutes with the interpretation of the function symbols over Σ , follows from the fact that f is by the injectivity of J k . Concerning thepreservation of atomic relations over Σ ′ , this is guaranteed by the fact that,were they not preserved, f would preserve the relations of satisfaction byelements of the geometric formulae over Σ defining the complements of suchrelations, contradicting the fact that f is a Σ ′ -structure homomorphism. (cid:3) A useful sufficient condition for condition ( iii )( b ) - (1) of Theorem 5.1 tohold, which is applicable to categories K whose arrows are not necessarilymonomorphisms is the following. Theorem 6.14.
Let T be a geometric theory over a signature Σ , and K asmall full subcategory of T -mod ( Set ) such that every model in K is a quotientof a T -model which is finitely presented by a geometric formula over Σ . Then T satisfies condition ( iii )( b ) - (1) of Theorem 5.1 with respect to the category K . Proof
We have to show that the extension functor u T ( K , E ) : Flat ( K op , E ) → Flat J T ( C T , E ) of section 3.3 is faithful.By Theorem 3.5, for any formula { ~x . φ } presenting a T -model M { ~x.φ } , u T ( K , E ) ( F )( { ~x . φ } ) ∼ = F ( M { ~x.φ } ) . The thesis thus follows immediately fromProposition 6.17. (cid:3) .3 Reformulations of condition ( iii ) of Theorem 5.1 Let us work in the context of a faithful interpretation of theories as in thelast section.Let u T ( K , E ) : Flat ( K op , E ) → T -mod ( E ) and u T ′ ( H , E ) : Flat ( H op , E ) → T ′ -mod ( E ) be the functors of section 3.3.Suppose that T ′ is of presheaf type and that H contains a category P whose Cauchy-completion coincides with the category of finitely presentable T ′ -models (so that T ′ is classified by the topos [ P , Set ] ). Notice that, sincethe functor f : K → H is a restriction of the functor a Set , K gets identifiedunder it with a subcategory of H and hence, by Remark 3.2(c), the extensionfunctor ξ E : Flat ( K op , E ) → Flat ( H op , E ) along the geometric morphism [ f, Set ] is faithful.From the fact that T ′ faithfully interprets in T (in the sense of Definition6.1) it follows that the functor u T ( K , E ) is faithful (resp. full and faithful) ifand only if the composite functor u T ′ ( H , E ) ◦ ξ E is, when regarded as a functorwith values in the subcategory Im ( a E ) of T ′ -mod ( E ) . The interest of thisreformulation lies in the alternative description of the functor u T ′ ( H , E ) : Flat ( H op , E ) → T ′ -mod ( E ) which is available under our hypotheses. Indeed, the following propositionholds. Proposition 6.15.
Under the hypotheses specified above, the functor u T ′ ( H , E ) : Flat ( H op , E ) → T ′ -mod ( E ) ≃ Flat ( P op , E ) sends any flat functor F : H op → E to its restriction F |P : P op → E , andacts accordingly on the natural transformations. Proof
This immediately follows from Theorem 3.5 in view of the fact that H contains P and all the objects of P are finitely presented T ′ -models (since T ′ is by our hypothesis of presheaf type classified by the topos [ P , Set ] ). (cid:3) Theorem 6.16.
Under the hypotheses specified above, we have that(i) T satisfies condition ( iii )( b ) - (1) of Theorem 5.1 with respect to the cat-egory K if and only if for any Grothendieck topos E , every naturaltransformation between flat functors in the image of the functor ξ E isdetermined by its restriction to the category P ;(ii) T satisfies condition ( iii )( b ) - (2) of Theorem 5.1 with respect to the cat-egory K if and only if for any Grothendieck topos E , every naturaltransformation between restrictions to P of flat functors in the imageof the functor ξ E can be extended to a natural transformation betweenthe functors themselves. Let us now consider the satisfaction of condition ( iii )( b ) of Theorem 5.1by the injectivization of a geometric theory T satisfying the hypotheses ofPropositions 6.11 and 6.13. Since by the proposition every finitely generated T -model is finitely presentable in T -mod ( Set ) and every model of T is adirected union of its finitely generated submodels, the category T -mod ( Set ) is equivalent to Flat ( f.g. T -mod ( Set ) op , Set ) , where f.g. T -mod ( Set ) is thefull subcategory of T -mod ( Set ) on the finitely generated T -models. Thefollowing theorem shows that condition ( iii )( b ) - (1) is satisfied. Before statingit, we need a lemma. Lemma 6.17.
Let C be a small category, E a topos and F : C op → E a flatfunctor. Then for any epimorphism f : a → b in C , the arrow F ( f ) : F ( b ) → F ( a ) is monic in E . Proof
The thesis follows at once from the fact that flat functors preserveall the finite limits which exist in the domain category using the well-knowncharacterization of monomorphisms in terms of pullbacks. (cid:3)
Theorem 6.18.
Let T be the injectivization of a theory of presheaf type T ′ (in the sense of Definition 6.33) such that the every finitely presentable T ′ -model is finitely generated (cf. for instance Proposition 6.11) and the finitelygenerated T ′ -models are all quotients (in the sense of section 6.2) of finitelypresentable T ′ -models (cf. for instance Proposition 6.12). Then T satisfiescondition ( iii )( b ) - (1) of Theorem 5.1 with respect to the category of finitelygenerated T ′ -models. Proof
It suffices to apply Proposition 6.15 and Lemma 6.17 in conjunctionwith Theorem 6.16 by taking K equal to the category of finitely generated T ′ -models and sortwise monic homomorphisms between them, H equal tothe category of finitely generated T ′ -models and homomorphisms betweenthem and P equal to the category of finitely presentable T ′ -models. (cid:3) ( ii ) of Theorem 6.16 holds. Proposition 6.19.
Let D be a full subcategory of a category C with filteredcolimits such that• for any object c of C there exists an epimorphism a → c in C from anobject a of D to c ,• every object of D is finitely presentable in C and• every object c of C can be canonically expressed as a filtered colimit colim ( d i ) of objects of D , with colimit arrows J i : d i → c (for i ∈ I ).Let F and G be flat functors C op → E . Then a natural transformation β : F | D → G | D lifts (uniquely) to a natural transformation F → G if andonly if for every epimorphism q : a → b of the form J i , the arrow β ( a ) : F ( a ) → G ( a ) restricts to an arrow F ( b ) → G ( b ) along the monomorphisms F ( q ) : F ( b ) → F ( a ) and G ( q ) : G ( b ) → G ( a ) . Proof
Clearly, the ‘only if’ direction is trivially satisfied, so we only haveto care about the ‘if’ direction. The uniqueness of the extended naturaltransformation follows from Proposition 6.17, so we just have to prove theexistence.Notice that for every object c of C there exists an i ∈ I such that thearrow J i : d i → c is an epimorphism. Indeed, by our hypotheses there existsan epimorphism q : a → c to c from an object a of D to c and, since a isfinitely presentable in C , q necessarily factors through an arrow of the form J i , which is necessarily epic as q is.Let β : F | D → G | D be a natural transformation. Choose an i ∈ I suchthat the arrow J i : d i → c is an epimorphism (such arrow exists by the aboveremark), and set α ( c ) equal to the restriction of β ( d i ) along the arrows F ( J i ) and G ( J i ) .First, let us verify that this definition is well-posed, i.e. that it does notdepend on the choice of i ∈ I . For any other index j such that J j is anepimorphism with codomain c , the filteredness of I ensures the existence ofan index k and two arrows ξ : i → k and χ : j → k in I , whence J k ◦ ξ = J i and J k ◦ χ = J j . Now, if we denote by u the restriction of the arrow β ( d k ) along the arrows F ( J k ) and G ( J k ) we obtain, by invoking the naturality of β with respect to the arrows ξ and χ in I , that u is equal to the uniquerestriction of β ( d i ) along the arrows F ( J i ) and G ( J i ) and to the uniquerestriction of β ( d j ) along the arrows F ( J j ) and G ( J j ) ; in particular, thesetwo restrictions are equal, as required.104t remains to show that the assignment c → α ( c ) is natural in c , i.e. thatfor any arrow f : c → c ′ in C , the following naturality square commutes: F ( c ) α ( c ) / / G ( c ) F ( c ′ ) F ( f ) O O α ( c ′ ) / / G ( c ′ ) . G ( f ) O O To prove this we observe that, using the fact that there exists an epimorphism J i : d i → c from an object of D to c , we can suppose without loss of generality c to lie in D ; indeed, since G sends epimorphisms to monomorphisms, thecommutativity of the naturality square above is equivalent, by definition of α ( c ) , to the commutativity of the naturality square relative to the arrow f ◦ J i . Now, consider the canonical representation of c ′ as a filtered colimit colim ( e j ) (for j ∈ J ) of objects in D , with colimit arrows K j : e j → c ′ ( j ∈ J ). Since c lies in D , c is finitely presentable in C by our hypothesesand hence there exists an index j ∈ J and an arrow r : c → e j suchthat K j ◦ r = f . The commutativity of our diagram thus follows from thecommutativity of the naturality square of β relative to the arrow r and thedefinition of α ( c ′ ) in terms of β ( e j ) . (cid:3) The following result, obtained by combining Theorem 6.45 and Proposi-tion 6.19, gives a criterion for condition ( iii )( b ) - (2) to hold. Theorem 6.20.
Let T be the injectivization of a theory of presheaf type T ′ such that every finitely presentable T ′ -model is finitely generated (cf. for in-stance Proposition 6.11), the finitely generated T ′ -models are all quotients(in the sense of section 6.2) of finitely presentable T ′ -models (cf. for in-stance Proposition 6.12) and every monic T ′ -model homomorphism betweenfinitely generated models of T is sortwise monic. Then T satisfies condition ( iii )( b ) - (2) of Theorem 5.1 with respect to the category C of finitely generatedmodels of T ′ and sortwise monic homomorphisms between them if and onlyif for any flat functors F, G : C op → E , denoting by ˜ F and ˜ G their extensionto the category of finitely generated T ′ -models and homomorphisms betweenthem and by P the full subcategory of C on the finitely presentable T ′ -models,any natural transformation α : F | P → G | P satisfies the following property:for any quotient T -model homomorphism q : a → b , where a is finitely pre-sentable and b is finitely generated, the arrow α ( b ) restricts along the monicarrows F ( q ) and G ( q ) . (cid:3) Remark 6.21.
Notice that if F and G are representable functors then thecondition of the theorem is trivially satisfied, the category T ′ -mod ( Set ) beingfinitely accessible. 105 .4 Quotient theories Recall from [9] that a quotient of a geometric theory T over a signature Σ isa geometric theory T ′ over Σ such that every axiom of T is provable in T ′ .It is often useful, in investigating whether a given geometric theory S isof presheaf type, to consider it in relation to a theory T of presheaf type ofwhich S is a quotient.The following result is a corollary of Theorem 6.3 and Proposition 6.5. Corollary 6.22.
Let T be a theory of presheaf type over a signature Σ and S be a quotient of T such that all the finitely presentable S -models are finitelypresentable as T -models (for instance, T can be the empty theory over a finitesignature and S can be any geometric theory over Σ whose finitely presentablemodels are all finite, cf. Theorem 6.4 [6]). Suppose moreover that for anyobject E of a Grothendieck topos E and any Σ -structure homomorphism x : c → Hom E ( E, M ) , where c is a finitely presentable T -model and M is a S -model in E , there exists an epimorphic family { e i : E i → E | i ∈ I } in E and for each i ∈ I a T -model homomorphism f i : c → c i , where c i isa finitely presentable S -model, and a Σ -structure homomorphism x i : c i → Hom E ( E i , M ) such that x i ◦ f i = Hom E ( e i , M ) ◦ x for all i ∈ I . Then S isof presheaf type. Proof
Condition ( iii ) of Theorem 5.1 satisfied by Proposition 5.11, whilethe fact that conditions ( i ) and ( ii ) of Theorem 5.1 are satisfied followsimmediately from Theorem 6.3 in view of Proposition 6.5 (take T equal to S , T ′ equal to T , K equal to the category of finitely presentable S -models and H equal to the category of finitely presentable T -models - the fact that R a is E -full is clear). (cid:3) We shall say that the family of homomorphisms x i in the statement ofthe corollary defines a ‘localization’ of the homomorphism x . Remarks 6.23. (a) Corollary 6.22 generalizes Theorem 3.6 [32], whose hy-potheses, which are expressed syntactically in terms of geometric logic,once interpreted in topos-theoretic semantics, are stronger than those ofCorollary 6.22 in the particular case where T is the empty theory overa finite signature and S is any geometric theory over Σ whose finitelypresentable models are all finite.(b) Theorem 6.28 follows as an immediate consequence of the corollary, ob-serving that if T and T ′ are theories as in the hypotheses of the theoremthen any T -model homomorphism having as codomain a T ′ -model alsohas as domain a T ′ -model whence the hypotheses of the corollary aretrivially satisfied. 106c) Assuming the first hypothesis of the corollary, the second is actuallynecessary for S to be of presheaf type. Indeed, if S is of presheaf typethen every model M of S is a E -filtered colimit of the associated canonicaldiagram of ‘constant’ finitely presentable S -models (cf. sections 4.4 and5.3), which by the first hypothesis are also finitely presentable T -models;the E -finite presentability of c thus implies the second hypothesis of thecorollary (cf. Proposition 4.15).(d) Corollary 6.22 is often applied to pairs of the form ( S , T = S c ) , where S isa geometric theory over a signature Σ and T equal to the cartesianization (or finite-limit part) of S c of S , namely the cartesian theory over Σ consisting of all the cartesian sequents over Σ which are provable in S (recall that a S -cartesian sequent is a sequent over Σ involving S -cartesian formulae over Σ , that is formulae built from atomic formulaeby only using finitary conjunctions and S -provably unique existentialquantifications). Notice that the cartesianization of S is the biggestcartesian theory over the signature of S of which S is a quotient.The question of whether for any theory of presheaf type T , every finitelypresentable T -model is finitely presentable as a T c -model is still open andwill be addressed in full generality in another paper. For the moment,we limit ourselves to remarking that this property is satisfied by all theexamples of theories of presheaf type considered in this paper (cf. section8).Corollary 6.22 can also be applied to pairs consisting of a geometrictheory over a signature Σ and the empty theory over the same signature,provided that the former satisfies appropriate hypotheses. In order to applythe corollary in this context, we need the following lemma. Below, when wesay that a Σ -structure c is finite we mean that for every sort A over Σ , cA is finite. Lemma 6.24.
Let T be a geometric theory over a finite signature Σ with afinite number of axioms each of which is of the form ( ⊤ ⊢ ~x W i ∈ I φ i ) , where the φ i are finite conjunctions of atomic formulae. Then for any T -model M in aGrothendieck topos E , any object E of E and any Σ -structure homomorphism f : c → Hom E ( E, M ) from a finite Σ -structure c there exists an epimorphicfamily { e i : E i → E | i ∈ I } in E and for each i ∈ I a finite Σ -substructure c i of Hom E ( E i , M ) which is a model of T and a Σ -structure homomorphism f i : c → c i such that Hom E ( e i , M ) ◦ f = j i ◦ f i (where j i is the canonicalinclusion of c i into Hom E ( E i , M ) ). Proof
For each axiom σ of the form ( ⊤ ⊢ ~x σ W i ∈ I σ φ σi ) and any finite tuple ~ξ of elements of c of the same type as ~x σ , since M is by our hypotheses a107 -model, there exists an epimorphic family { e ~ξi : E ~ξi → E | i ∈ I σ } in E suchthat for any i ∈ I σ , f ( ~ξ ) ◦ ~e ξi factors through [[ ~x σ . φ i ]] M . Since there isonly a finite number of axioms of T and of elements of c , and the fiberedproduct of a finite number of epimorphic families is again an epimorphicfamily, there exists an epimorphic family { e k : E k → E | k ∈ K } such thatfor any k ∈ K , any axiom σ of T and any tuple ~ξ of elements of c of theappropriate sorts, the element f ( ~ξ ) ◦ e k ∈ Hom E ( E k , M ) satisfies the formulaon the right-hand side of the sequent σ . Hence if we consider, for each k ∈ K , the surjection-inclusion factorization i k ◦ f k of the homomorphism Hom E ( e k , M ) ◦ f : c → Hom E ( E k , M ) (in the sense of Lemma 6.51), weobtain that c k is finite, since f k is surjective, and that all the axioms of T are satisfied in c k (since the formulae φ i are finite conjunctions of atomicformulae and c k is a substructure of Hom E ( E k , M ) ), that is all the c k aremodels of T . This proves our thesis. (cid:3) Corollary 6.25.
Let T be a geometric theory over a finite signature Σ with afinite number of axioms each of which is of the form ( ⊤ ⊢ ~x W i ∈ I φ i ) , where the φ i are finite conjunctions of atomic formulae. Suppose that for any T -model M in a Grothendieck topos E and object E of E , every finitely generated Σ -substructure of Hom E ( E, M ) has sortwise only a finite number of elementsbesides the constants (for instance, when the signature Σ does not containfunction symbols except for a finite number of constants). Then T is ofpresheaf type, classified by the category of covariant set-valued functors onthe category of finite models of T . Proof
As the signature of T is finite, every model which sortwise containsonly finitely many elements besides the constants is finitely presentable as amodel of the empty theory over the signature of T (cf. Theorem 6.4 [6]). Thisremark ensures, by Lemma 6.24, that the theory T satisfies the hypothesesof Corollary 6.22 with respect to the empty theory over its signature. Sincethe latter theory is clearly of presheaf type, it follows that T is of presheaftype as well, as required. (cid:3) In this section we shall analyze the presheaf-type quotients of a presheaf-typetheory T in terms of the associated subtoposes of the classifying topos for T .We know from [9] that every quotient T ′ a theory of presheaf type T corre-sponds to a unique Grothendieck topology J on the category f.p. T -mod ( Set ) op (under the duality theorem of [9]).Recall from [21] (Definition C2.2.18) that a Grothendieck topology J ona category C is said to be rigid if for every object c of C , the family of allthe arrows to c in C from J -irreducible objects of C (i.e., objects d with the108roperty that the only J -covering sieve on d is the maximal one), generatesa J -covering sieve.We can characterize the topologies J such that the corresponding subto-pos Sh ( f.p. T -mod ( Set ) op , J ) ֒ → [ f.p. T -mod ( Set ) , Set ] is an essential geo-metric inclusion (that is, a geometric inclusion whose inverse image whichadmits a left adjoint) by using the following site characterizations: (1) A canonical geometric inclusion Sh ( C , J ) ֒ → [ C op , Set ] from a topos Sh ( C , J ) which is equivalent to a presheaf topos and such that C is a Cauchy-complete category, is essential if and only if the topology J is rigid; (2) A geometric morphism (resp. geometric inclusion) [ C , Set ] → [ D , Set ] ,where D is a Cauchy-complete category, is essential if and only if it is inducedby a functor (resp. by a full and faithful functor) C → D .The latter characterization is well-known (cf. Lemma A4.1.5 [21] andExample A4.2.12(b)), while the former can be proved as follows. If J isrigid then, denoting by D the full subcategory of C on the J -irreducible ob-jects, the Comparison Lemma yields an equivalence Sh ( C , J ) ≃ [ D op , Set ] which makes the canonical inclusion Sh ( C , J ) ֒ → [ C op , Set ] isomorphic tothe canonical geometric inclusion [ D op , Set ] ֒ → [ C op , Set ] induced by the fullinclusion of categories D op ֒ → C op ; in particular, the morphism Sh ( C , J ) ֒ → [ C op , Set ] is essential. Conversely, if the canonical geometric inclusion i J : Sh ( C , J ) ֒ → [ C op , Set ] is essential then if Sh ( C , J ) is equivalent to [ D op , Set ] ,by property (2) i J is isomorphic to a geometric inclusion [ D op , Set ] → [ C op , Set ] induced by a full embedding D op ֒ → C op . Now, the topology J D on C defined by saying that a sieve R on c is J -covering if and only ifit contains all the morphisms from objects of D to c is clearly rigid, andthe Comparison Lemma yields an equivalence Sh ( C , J D ) ≃ [ D op , Set ] whichmakes the geometric morphism [ D op , Set ] → [ C op , Set ] isomorphic to thecanonical inclusion Sh ( C , J D ) ֒ → [ C op , Set ] (cf. Example A2.2.4(d) [21]). Itthus follows that J = J D ; in particular, J is rigid.These site characterizations define the arches of a ‘bridge’ (in the senseof [13]) leading to the following result (cf. also Theorem 6.8 [6] for a relatedresult). Theorem 6.26.
Let T ′ be a quotient of a theory of presheaf type T , cor-responding to a Grothendieck topology J on the category f.p. T -mod ( Set ) op under the duality theorem of [9]. Suppose that T ′ is itself of presheaf type.Then every finitely presentable T ′ -model is finitely presentable also as a T -model if and only if the topology J is rigid. Proof
If every finitely presentable T ′ -model is finitely presentable also asa T -model then the geometric inclusion corresponding via the duality the-orem to the quotient T ′ of T is induced by the inclusion of the respectivefull subcategory of f.p. T -mod ( Set ) on the T ′ -models. In view of the abovecharacterizations, the topology J is rigid. Conversely, if J is rigid then109 h ( f.p. T -mod ( Set ) op , J ) is equivalent to the presheaf topos [ D , Set ] , where D op is the full subcategory of f.p. T -mod ( Set ) op on the J -irreducible objects.Clearly, this subcategory is Cauchy-complete (as f.p. T -mod ( Set ) is Cauchy-complete and the condition of J -irreducibility is stable under retracts) andhence, since T ′ is of presheaf type, it is equivalent to the category of finitelypresentable T ′ -models; in particular, any such model is finitely presentableas a T -model. (cid:3) Remark 6.27. (i) Under the hypothesis that every finitely presentable T ′ -model is finitely presentable as a T -model, the full subcategoryf.p. T ′ -mod ( Set ) of f.p. T -mod ( Set ) and the topology J can be definedin terms of each other as follows (cf. the proof of the theorem): theobjects of f.p. T ′ -mod ( Set ) are precisely the J -irreducible objects off.p. T -mod ( Set ) , and a sieve S in f.p. T ′ -mod ( Set ) on an object c is J -covering if and only if it contains all the arrows in f.p. T -mod ( Set ) from finitely presentable T ′ -models to c .(ii) If T ′ is a presheaf-type quotient of a theory of presheaf type T then forany finitely presentable T -model c , any T ′ -model M and any T -modelhomomorphism f : c → M , there exists a T -model homomorphism g : c → c ′ to a finitely presentable T ′ -model c ′ and a T -model homo-morphism h : c ′ → M such that h ◦ g = f . Indeed, M can be expressedas a filtered colimit of finitely presentable T ′ -models c ′ ; therefore, as c is finitely presentable as a T -model, f necessarily factors through acolimit arrow c ′ → M .(iii) Under the hypothesis that every finitely presentable T ′ -model is finitelypresentable as a T -model, by the duality theorem of [9], the syntacticdescription of T ′ in terms of J given therein and the description of J in terms of f.p. T ′ -mod ( Set ) given in Remark 6.27(a), T ′ can be charac-terized as the quotient of T obtained by adding all the sequents of theform ( φ ( ~x ) ⊢ ~x W i ∈ I ∃ ~x i θ i ( ~x i , ~x )) , where { θ i ( ~x i , ~x ) : φ i ( ~x i ) → φ ( ~x ) } is thefamily of T -provably functional formulae from the formulae φ i ( ~x i ) pre-senting a T ′ -model to a T -irreducible formula (equivalently, a formulapresenting a T -model) φ ( ~x ) .The following result provides a natural class of presheaf-type quotientsof presheaf-type theories whose associated topologies are rigid. Theorem 6.28.
Let T be a theory of presheaf type over a signature Σ . Thenany quotient T ′ of T obtained from T by adding sequents of the form φ ⊢ ~x ⊥ ,where φ ( ~x ) is a geometric formula over Σ , is classified by the topos [ T , Set ] ,where T is the full subcategory of f.p. T -mod ( Set ) on the T ′ -models. Proof
By covering each φ ( ~x ) by T -irreducible formulae in the syntacticcategory C T of T , we can suppose without loss of generality all of the φ ( ~x ) to110e T -irreducible, that is to present a T -model M φ . Then, by the results of[9], the quotient T ′ is classified by the topos Sh ( f.p. T -mod ( Set ) op , J ) , where J is the smallest Grothendieck topology on f.p. T -mod ( Set ) op which containsall the empty sieves on the models M φ presented by the formulae φ involvedin the axioms φ ⊢ ~x ⊥ added to T to form T ′ . Let T be the full subcategoryof f.p. T -mod ( Set ) on the T ′ -models. Then T is J -dense; indeed, any objectnot in T admits an arrow from a model of the form M φ and hence is coveredby the empty sieve. Further, for any object of T , the J -covering sieves on itare exactly the maximal ones; therefore, the Comparison Lemma yields anequivalence Sh ( f.p. T -mod ( Set ) op , J ) ≃ [ T , Set ] , as required. (cid:3) The following theorem provides a method for constructing theories of presheaftype whose categories of finitely presentable models are equivalent, up toCauchy-completion, to a given small category of structures.
Theorem 6.29.
Let T be a theory of presheaf type and A a full subcategory off.p. T -mod ( Set ) . Then the A -completion T ′ of T (i.e., the set of all geometricsequents over the signature of T which are valid in all models in A ) is ofpresheaf type classified by the topos [ A , Set ] ; in particular, every finitelypresentable T ′ -model is a retract of a model in A . Proof
Since every model in A is finitely presentable as a T -model, we havea geometric inclusion i : [ A , Set ] ֒ → Set [ T ] ≃ [ f.p. T -mod ( Set ) , Set ] inducedby the canonical inclusion A ֒ → f.p. T -mod ( Set ) . This subtopos corresponds,by the duality theorem of [9], to a quotient T ′ of T classified by the topos [ A , Set ] , which can be characterized as the collection of all geometric se-quents which hold in every model in A , that is, as the A -completion T ′ of T (recall that theories of presheaf type have enough set-based models). There-fore T ′ is of presheaf type classified by the topos [ A , Set ] ; but the finitelypresentable T ′ -models are the finitely presentable objects of Ind - A , that isthe retracts of objects of A in Ind - A ≃ T ′ -mod ( Set ) . This completes theproof of the theorem. (cid:3) Remarks 6.30. (a) Theorem 6.29 is a generalization of Joyal and Wraith’srecognition theorem (Theorem 1.1 [2]), as well as of Proposition 5.3 [22].Indeed, the former theorem can be obtained as the particular case ofTheorem 6.29 when T ′ has enough set-based models and every set-basedmodel of T ′ is a filtered colimit of finitely presentable models of T ′ whichare also finitely presentable as T -models, while Proposition 5.3 [22] canbe obtained as the specialization of this latter theorem to the case when T ′ is a disjunctive theory and T is a cartesian theory (in view of the fact,proved in [22], that the category of set-based models of a disjunctive111heory is multiply finitely presentable and hence that every set-basedmodel of a disjunctive theory can be expressed as a filtered colimit offinitely presentable models of it).(b) It is worth to compare Theorem 6.29 with Theorem 5.1. If T ′ is a quotientof a theory of presheaf type T and K is a small category of set-based T ′ -models such that every model in K is finitely presentable as a T -modelthen, in order to conclude that T ′ is of presheaf type classified by thetopos [ K , Set ] , we can either verify conditions ( i ) and ( ii ) of Theorem5.1 or to verify that the models in K are jointly conservative for T ′ ; infact, as observed in Remark 5.2(c), conditions ( i ) and ( ii ) of Theorem5.1 together imply that the models in K are jointly conservative for T ′ . Corollary 6.31.
Assuming the axiom of choice, every coherent theory (or,more generally, any theory in a countable fragment of geometric logic as inthe hypotheses of Theorem 5.1.7 [27]) T over a finite relational signaturewhose axioms do not contain existential quantifications is of presheaf type. Proof
Since the axioms of T do not contain quantifications, every substruc-ture of a model of T is a model of T . Moreover, since the signature of T is relational, every finitely generated substructure over the signature of T contains only a finite number of elements besides the constants. Thereforeevery finitely presentable model of T contains only a finite number of ele-ments besides the constants (as any model of T is a filtered union of its finitesubmodels); on the other hand, since the signature of T is finite, every suchmodel is finitely presentable as a model of the empty theory over the signa-ture of T (cf. Theorem 6.4 [6]). Therefore condition ( iii ) of Theorem 5.1is satisfied (cf. Proposition 5.11(i)). Now, under the axiom of choice, everytheory satisfying the hypotheses of Theorem 5.1.7 [27], in particular any co-herent theory, has enough models, whence our thesis follows from Theorem6.29. (cid:3) It is nonetheless important, from a constructive viewpoint, to be able toprove that a theory is of presheaf type without invoking the axiom of choice.Theorem 5.1 allows us to do so in a great variety of cases.Theorem 6.29 shows that a good first step in constructing a geometrictheory classified by a presheaf topos [ K , Set ] consists in finding a theory ofpresheaf type T such that the category K can be identified as a full sub-category of the category of finitely presentable models of T . Indeed, underthese hypotheses, Theorem 6.29 ensures the existence of a quotient T K of T classified by the topos [ K , Set ] , which can be characterized as the theoryconsisting of all the geometric sequents over the signature of T which arevalid in every model in K . In most cases, if one has a natural candidate S for a theory classified by the topos [ K , Set ] , the theory T can be chosen tobe the Horn part of S or the cartesianization S c of S .112f course, the abstract characterization of T K as the theory consisting ofall the geometric sequents over the signature of T which are valid in everymodel in K is not very useful in specific contexts, where one looks for anaxiomatization of T K as simple and ‘economical’ as possible. To this end, weobserve that if every M in K is strongly finitely presented as a model of T aswell as finitely generated (in the sense that for any sort A over the signatureof T the elements of the set M A are precisely given by the interpretationsin M of terms t A ( ~x ) (or more generally of T -provably functional predicates)over the signature of T , where ~x are the generators of M as strongly finitelypresented model of T ) then we dispose of an explicit axiomatization of thetheory T K , as given by the following Theorem 6.32.
Let T be a geometric theory over a signature Σ and K a fullsubcategory of the category set-based T -models such that every T -model in K is both strongly finitely presentable and finitely generated (with respect to thesame generators). Then the following sequents (where we denote by P the setof geometric formulae over Σ which strongly present a T -model in K ), addedto the axioms of T , yield an axiomatization of a quotient of T classified by thetopos [ K , Set ] via a Morita-equivalence induced by the canonical geometricmorphism [ K , Set ] → Sh ( C T , J T ) .In particular, if the theory T is of presheaf type (whence every finitelypresentable T -model is strongly finitely presentable) and all the models in thefull subcategory K of T -mod ( Set ) are finitely presented and finitely gener-ated (with respect to the same generators), the following sequents yield anaxiomatization of the theory T K defined above:(i) The sequent ( ⊤ ⊢ [] _ φ ( ~x ) ∈P ( ∃ ~x ) φ ( ~x )); (ii) For any formulae φ ( ~x ) and ψ ( ~y ) in P , where ~x = ( x A , . . . , x A n n ) and ~y = ( y B , . . . , y B m m ) , the sequent ( φ ( ~x ) ∧ ψ ( ~y ) ⊢ ~x,~y _ χ ( ~z ) ∈P ,t A ( ~z ) ,...,t Ann ( ~z ) s B ( ~z ) ,...,s Bmm ( ~z ) ( ∃ ~z )( χ ( ~z ) ∧ ^ i ∈{ ,...,n } ,j ∈{ ,...,m } ( x i = t i ( ~z ) ∧ y j = s j ( ~z )))) , where the disjunction is taken over all the formulae χ ( ~z ) in P and allthe sequences of terms t A ( ~z ) , . . . , t A n n ( ~z ) and s B ( ~z ) , . . . , s B m m ( ~z ) whoseoutput sorts are respectively A , . . . , A n , B , . . . , B m and such that, de-noting by ~ξ the set of generators of the model M { ~z.χ } (strongly) finitelypresented by the formula χ ( ~z ) , ( t A ( ~ξ ) , . . . , t A n n ( ~ξ )) ∈ [[ ~x . φ ]] M { ~z.χ } and ( s B ( ~ξ ) , . . . , s B m m ( ~ξ )) ∈ [[ ~y . ψ ]] M { ~z.χ } ; iii) For any formulae φ ( ~x ) and ψ ( ~y ) in P , where ~x = ( x A , . . . , x A n n ) and ~y = ( y B , . . . , y B m m ) , and any terms t A ( ~y ) , s A ( ~y ) , . . . , t A n n ( ~y ) , s A n n ( ~y ) whose output sorts are respectively A , . . . , A n , the sequent ( ^ i ∈{ ,...,n } ( t i ( ~y ) = s i ( ~y )) ∧ φ ( t /x , . . . , t n /x n ) ∧ φ ( s /x , . . . , s n /x n ) ∧ ψ ( ~y ) ⊢ ~y _ χ ( ~z ) ∈P ,u B ( ~z ) ,...,u Bmm ( ~z ) (( ∃ ~z )( χ ( ~z ) ∧ ^ j ∈{ ,...,m } ( y j = u j ( ~z ))) , where the disjunction is taken over all the formulae χ ( ~z ) in P andall the sequences of terms u B ( ~z ) , . . . , u B m m ( ~z ) whose output sorts arerespectively B , . . . , B m and such that, denoting by ~ξ the set of gener-ators of the model M { ~z.χ } (strongly) finitely presented by the formula χ ( ~z ) , ( u B ( ~ξ ) , . . . , u B m m ( ~ξ )) ∈ [[ ~y . ψ ]] M { ~z.χ } and t i ( u ( ~ξ ) , . . . , u m ( ~ξ )) = s i ( u ( ~ξ ) , . . . , u m ( ~ξ )) in M { ~z.χ } for all i ∈ { , . . . , n } ;(iv) For any sort A over Σ , the sequent ( ⊤ ⊢ x A _ χ ( ~z ) ∈P ,t A ( ~z ) ( ∃ ~z )( χ ( ~z ) ∧ x = t ( ~z ))) , where the the disjunction is taken over all the formulae χ ( ~z ) in P andall the terms t A ( ~z ) whose output sort is A ;(v) For any sort A over Σ , any formulae φ ( ~x ) and ψ ( ~y ) in P , where ~x = ( x A , . . . , x A n n ) and ~y = ( y B , . . . , y B m m ) , and any terms t A ( ~x ) and s A ( ~y ) , the sequent ( φ ( ~x ) ∧ ψ ( ~y ) ∧ t ( ~x ) = s ( ~y ) ⊢ ~x,~y _ χ ( ~z ) ∈P ,p A ( ~z ) ,...,p Ann ( ~z ) q B ( ~z ) ,...,q Bmm ( ~z ) ( ∃ ~z )( χ ( ~z ) ∧∧ ^ i ∈{ ,...,n } ,j ∈{ ,...,m } ( x i = p i ( ~z ) ∧ y j = q j ( ~z ))) , where the disjunction is taken over all the formulae χ ( ~z ) in P and allthe sequences of terms p A ( ~z ) , . . . , p A n n and q B ( ~z ) , . . . , q B m m ( ~z ) whoseoutput sorts are respectively A , . . . , A n , B , . . . , B m and such that, de-noting by ~ξ the set of generators of the model M { ~z.χ } (strongly) finitelypresented by the formula χ ( ~z ) , ( p A ( ~ξ ) , . . . , p A n n ( ~ξ )) ∈ [[ ~x . φ ]] M { ~z.χ } and ( q B ( ~ξ ) , . . . , q B m m ( ~ξ )) ∈ [[ ~y . ψ ]] M { ~z.χ } and t ( p ( ~ξ ) , . . . , p n ( ~ξ )) = s ( q ( ~ξ ) , . . . , q m ( ~ξ )) in M { ~z.χ } . roof Let R be the geometric theory obtained from T by adding all thesequents specified above. The objects in K are clearly models of R (cf.Remarks 5.6(a) and 5.10). From the fact that they are strongly finitelypresentable it follows by Proposition 5.11(ii) that condition ( iii ) of Theorem5.1 is satisfied by the theory R with respect to the category K . By Theorems5.3 and 5.9 and Remarks 5.6(a) and 5.10, R also satisfies conditions ( i ) and ( ii ) of Theorem 5.1 with respect to K . Theorem 5.1 thus implies that R is of presheaf type classified by the topos [ K , Set ] . Moreover, it can bereadily seen that the resulting Morita-equivalence is induced by the canonicalgeometric morphism from [ K , Set ] to the classifying topos for T ; this ensuresin particular that if T is of presheaf type then R is equal to the theory T K . (cid:3) For any geometric theory T , we can slightly modify its syntax so to obtain ageometric theory whose models in Set are the same as those of T and whosehomomorphisms between them are precisely the sortwise injective T -modelhomomorphisms.This construction is useful in many contexts. For instance, in [9] weshowed that if the category f.p. T -mod ( Set ) of finitely presentable modelsof a theory of presheaf type T satisfies the amalgamation property thenthe quotient of T corresponding to the subtopos Sh ( f.p. T -mod ( Set ) op , J at ) of [ f.p. T -mod ( Set ) , Set ] (where J at is the atomic topology) via the dualitytheorem of [9] axiomatizes the homogeneous T -models (in the sense of [5]) inany Grothendieck topos. Now, the notion of homogeneous T -model, whichis strictly related to the notion of weakly homogeneous model considered inclassical Model Theory (cf. [20]), is mostly interesting when the arrows ofthe category T -mod ( Set ) are all monic; indeed, as shown in [6], a necessarycondition for T to admit an associated ‘concrete’ Galois theory, is that allthe arrows in f.p. T -mod ( Set ) should be strict monomorphisms.This motivates the following formal definition. Definition 6.33.
Let T be a geometric theory over a signature Σ . The injectivization T m of T is the geometric theory obtained from T by addinga binary predicate D A A, A for each sort A over Σ and the coherentsequents ( D A ( x A , y A ) ∧ x A = y A ) ⊢ x A ,y A ⊥ ) and ( ⊤ ⊢ x A ,y A D A ( x A , y A ) ∨ x A = y A ) . The models of T m in an arbitrary topos E coincide with the models M of T in E which are sortwise decidable, in the sense that for every sort A over the signature of T the object M A of E is decidable , i.e. the diagonalsubobject of M A is complemented (by the interpretation of D A in M ).115s shown by the following lemma, the arrows M → N in the category T m -mod ( Set ) are precisely the T -model homomorphisms f : M → N suchthat for every sort A over the signature of T , f A : M A → N A is a monomor-phism in E . Lemma 6.34.
Let A and B be decidable objects in a topos E and f : A → B an arrow in E . Let D A A × A and D B B × B denote respectively thecomplements of the diagonal subobjects δ A : A A × A and δ B : B B × B .Then f is a monomorphism if and only if f × f : A × A → B × B restrictsto an arrow D A → D B . Proof
It is immediate to see that f : A → B is a monomorphism if andonly if the diagram A δ A (cid:15) (cid:15) f / / B δ B (cid:15) (cid:15) A × A f × f / / B × B, is a pullback.Since pullback functors preserve arbitrary unions and intersections ofsubobjects in a topos (they having both a left and a right adjoint), we havethat ( f × f ) ∗ ( D B ) ∼ = ¬ ( f × f ) ∗ ( δ B ) . Now, f × f : A × A → B × B restrictsto an arrow D A → D B if and only if D A ≤ ( f × f ) ∗ ( D B ) . But this conditionholds if and only if D A ∩ ( f × f ) ∗ ( δ B ) ∼ = 0 , i.e. if and only if ( f × f ) ∗ ( δ B ) ≤ δ A ,which is equivalent to the condition ( f × f ) ∗ ( δ B ) ∼ = δ A (as δ A ≤ ( f × f ) ∗ ( δ B ) ). (cid:3) Several injectivizations of theories of presheaf type have been consideredin the literature, e.g. in [26] and in [21]; see also [7] and [6] for some appli-cations of this type of theories in the context of topos-theoretic Galois-typeequivalences.As we shall see below, under certain conditions, the injectivization of atheory of presheaf type is again of presheaf type.The following proposition is a corollary of Theorem 4.3.
Proposition 6.35.
Let T be a geometric theory. Then for any T -models M and N in a Grothendieck topos E which are sortwise decidable there exists anobject Hom E T m -mod ( E ) ( M, N ) of E satisfying the following universal property:for any object E of E we have an equivalence Hom E ( E, Hom E T m -mod ( E ) ( M, N )) ∼ = Hom T m -mod ( E /E ) (! ∗ E ( M ) , ! ∗ E ( N )) natural in E ∈ E . Remarks 6.36. (a) For any E , M and N , Hom E T m -mod ( E ) ( M, N ) embedscanonically as a subobject of Hom E T -mod ( E ) ( M, N ) .116b) Let T be a geometric theory over a signature Σ , c a finitely presentable T -model and M a sortwise decidable model of T in a Grothendieck topos E . Then the subobject Hom E T m -mod ( E ) ( γ ∗E ( c ) , M ) Hom E T -mod ( E ) ( γ ∗E ( c ) , M ) can be identified with the interpretation of the formula χ c := ^ A sort over Σ ,x,y ∈ cA, x = y ( π A ( f ( γ ∗E ( x )) , π A ( f ( γ ∗E ( y ))) ∈ D MA , written in the internal language of the topos E , where π A : Hom E T -mod ( E ) ( M, N ) → N A MA is the arrow defined in Remark 4.4(b) and x, y : 1 → c are the arrows in Set corresponding respectively to the elements x and y of cA . Indeed, a T -model homomorphism f : γ ∗E ( c ) → M is sortwise monic if and only iffor every sort A over Σ and any distinct elements x, y ∈ cA , f A ( γ ∗E ( x )) and f A ( γ ∗E ( y )) are disjoint, in other words they satisfy the relation D MA .It follows that if for every finitely presentable T -model c , the formula χ c is equivalent to a geometric formula over the signature of T m , forinstance when Σ only contains a finite number of sorts and for any sort A and any finitely presentable T -model the set c A is finite, then thetheory T m satisfies condition ( iii )( a ) of Theorem 5.1 with respect to thecategory of finitely presentable T -models if T does.(c) Let T be a geometric theory over a signature Σ , c a finitely presentable T -model and M a sortwise decidable model of T in a Grothendieck topos E .If c is strongly finitely presented by a geometric formula φ ( ~x ) over Σ thenthe subobject Hom E T -mod m ( E ) ( γ ∗E ( c ) , M ) of Hom E T -mod ( E ) ( γ ∗E ( c ) , M ) ∼ = [[ ~x .φ ]] M can be identified with the intersection of the interpretations in M of the formulae of the form ¬ ( ∃ x A )( θ ( ~x, x A ) ∧ θ ( ~x, x A )) , where A is asort over Σ and θ , θ are T -provably functional formulae from { ~x . φ } to { x A . ⊤} . Lemma 6.37.
Let T be a geometric theory over a signature Σ , c a set-based T -model and M a T -model in a Grothendieck topos E . A general-ized element x : E → G M ( c ) = Hom E T m -mod ( E ) ( γ ∗E ( c ) , M ) can be identifiedwith a Σ -homomorphism ξ x : c → Hom E ( E, M ) which is sortwise disjunc-tive in the sense that for every sort A over Σ , the function ξ x A : cA → Hom E ( E, M A ) has the property that for any distinct elements z, w ∈ cA ,the arrows ξ x A ( z ) , ξ x A ( w ) : E → M A have equalizer zero in E . roof The thesis follows from Proposition 4.5, observing that, by Lemmas6.41 and 6.43, an arrow τ A : γ ∗E /E ( cA ) → ! ∗ E ( M A ) in E /E is monic in E /E if and only if the corresponding arrow ξ A : cA → Hom E ( E, M A ) satisfiesthe property that for any distinct elements z, w ∈ cA , the arrows ξ A ( z ) : E → M A and ξ A ( w ) : E → M A have equalizer zero (notice that two arrows s, t : ( a : A → E ) → ( b : B → E ) in E /E have equalizer zero in E /E if andonly if the arrows s : A → B and t : A → B have equalizer zero in E ). (cid:3) Remark 6.38.
Suppose that the formula { x A . ⊤} strongly presents a T -model F A . Then a Σ -structure homomorphism s : F A → Hom E ( E, M ) ,corresponding to a generalized element z : E → M A , is sortwise disjunc-tive if and only if for any sort B over Σ and any two T -provably inequiv-alent T -provably functional geometric formulae θ and θ from { x A . ⊤} to { x B . ⊤} , the generalized elements [[ θ ]] M ◦ z and [[ θ ]] M ◦ z are dis-joint. Indeed, from the proof of Theorem 4.1 we know that that for anysort B and T -provably functional geometric formula θ from { x A . ⊤} to { x B . ⊤} , the function [[ θ ]] F A : F A A = Hom C T ( { x A . ⊤} , { x A . ⊤} ) → F A B = Hom C T ( { x A . ⊤} , { x B . ⊤} ) coincides with [ θ ] ◦ − : Hom C T ( { x A . ⊤} , { x A . ⊤} ) → Hom C T ( { x A . ⊤} , { x B . ⊤} ) , while the generator u A of F A is precisely the identity arrow on { x A . ⊤} . Now, it is immediate to see thatfor any Σ -structure homomorphism s : F A → Hom E ( E, M ) and any sort B over Σ , the diagram F A A [ θ ] ◦− (cid:15) (cid:15) sA / / Hom E ( E, M A ) [[ θ ]] M ◦− (cid:15) (cid:15) F A B sB / / Hom E ( E, M B ) commutes, i.e. sA ( u A ) = sB ( θ ) . From these remarks, our claim immediatelyfollows.Note that if T is a universal Horn theory (in the sense of [3]) then wecan suppose without loss of generality θ and θ to be functional formulae ofthe form x B = t ( x A ) , where t is a term of type A → B over Σ (cf. Remark4.2(a)). In this section we shall establish a general result about injectivizations oftheories of presheaf type, namely that for any theory of presheaf type T suchthat all the monic arrows in the category f.p. T -mod ( Set ) are sortwise monic,the injectivization of T satisfies condition ( iii ) of Theorem 5.1 with respectto the category f.p. T -mod ( Set ) .Before proving this theorem, we need a series of preliminary results.118 emma 6.39. Let r : R A × A a subobject in a Grothendieck topos E , e : E → A × A an arrow in E and { f i : E i → E | i ∈ I } an epimorphicfamily in E . If for every i ∈ I the arrow e ◦ f i factors through r then e factorsthrough r . Proof
The arrow ` i ∈ I f i : ` i ∈ I E i → E induced by the universal property of thecoproduct is an epimorphism, since by our hypothesis the family { f i : E i → E | i ∈ I } is epimorphic. The factorizations b i : E i → R of the arrows e ◦ f i through r induce an arrow b := ` i ∈ I b i : ` i ∈ I E i → R , such that r ◦ b = e ◦ ` i ∈ I f i .Consider the epi-mono factorization b = h ◦ k in E of the arrow b , where k : ` i ∈ I E i ։ U and h : U R , and the epi-mono factorization e = m ◦ n of the arrow e in E , where n : E ։ T and m : T A . Clearly, the arrow e ◦ ` i ∈ I f i factorizes both as m ◦ ( n ◦ ` i ∈ I f i ) and as ( r ◦ h ) ◦ k . Now, as m , r ◦ h aremonic and n ◦ ` i ∈ I f i , k are epic, the uniqueness of the epi-mono factorizationof a given arrow in a topos implies that there exists an isomorphism i : T ∼ = U such that r ◦ h ◦ i = m and i ◦ n ◦ ` i ∈ I f i = k . The arrow h ◦ i ◦ n thus providesa factorization of e through r , as required. (cid:3) Lemma 6.40.
Let { f i : A i → B | i ∈ I } be a family of arrows in aGrothendieck topos E . Then the arrow ` i ∈ I f i → ` i ∈ I A i → B is monic ifand only if for every i ∈ I , f i is monic and for every i, i ′ ∈ I , either i = i ′ or the subobjects f i : A i B and f ′ i : A i ′ B are disjoint. Proof
The ‘if’ direction follows from Proposition IV 7.6 [26], while the ‘onlyif’ one follows from the fact that coproducts in a topos are always disjoint(cf. Corollary IV 10.5 [26]) and the composite of a given monomorphismwith two disjoint subobjects yields two disjoint subobjects. (cid:3)
Remark 6.41.
If in the statement of Lemma 6.40 the objects A i are allequal to the terminal object E of E then the arrows f i are automaticallymonic and any two of them are disjoint if and only if their equalizer is zero.The notation employed in the statements and proofs of the followingresults is borrowed from section 3.2. Lemma 6.42.
Let F : D op → E be a flat functor from a subcategory D of asmall category C to a Grothendieck topos E and x : E → ˜ F ( d ) a generalizedelement which factors through χ Fd : F ( d ) → ˜ F ( d ) as χ Fd ◦ x ′ . Then the naturaltransformation α x : γ ∗E /E ◦ y C d → ! ∗ E ◦ ˜ F corresponding to x is equal to ˜ α x ′ ,where α x ′ is the natural transformation γ ∗E /E ◦ y D d → ! ∗ E ◦ F corresponding to x ′ . Proof
Straightforward from the results of section 3.2. (cid:3) C , we shall write C m for the category whose objectsare the objects of C and whose arrows are the monic arrows in C betweenthem. The following results concern extensions ˜ F of flat functors F alongthe embedding C m ֒ → C .Below, for a given flat functor H : C op → E with values in a Grothendiecktopos E and any object E of E , we write H E for the flat functor ! ∗ E ◦ H : C op → E /E . Proposition 6.43.
Let C be a small category and and F : C m op → E a flatfunctor. Then for any object c ∈ C and any generalized element x : E → ˜ F ( c ) , x factors through χ Fc : F ( c ) → ˜ F ( c ) if and only if the correspondingnatural transformation γ ∗E /E ◦ y C c → ˜! ∗ E ◦ F is pointwise monic. Proof
The ‘only if’ direction follows at once from Lemma 6.42 and Theorem6.8. To prove the ‘if’ one, thanks to the localization technique, we cansuppose without loss of generality E = 1 E .Suppose that the natural transformation α x : γ ∗E ◦ y C c → ˜ F correspondingto the generalized element x : 1 → ˜ F ( c ) is pointwise monic. We wantto prove that x factors through χ Fc : F ( c ) → ˜ F ( c ) . Recall that α x ( d ) : ` f ∈ Hom C ( d,c ) E → ˜ F ( c ) is defined as the arrow which sends the component ofthe coproduct indexed by f to the generalized element ˜ F ( f ) ◦ x : 1 → ˜ F ( d ) .By Lemma 6.40, if α x is pointwise monic then for any object d ∈ C andarrows f, g : d → c in C , either f = g or the equalizer of ˜ F ( f ) ◦ x and ˜ F ( g ) ◦ x is zero. Consider the pullbacks of the jointly epimorphic arrows κ ( a,z ) : F ( a ) → ˜ F ( c ) along the arrow x : 1 E → ˜ F ( c ) : E ( a,z ) h ( a,z ) / / e ( a,z ) (cid:15) (cid:15) F ( a ) κ ( a,z ) (cid:15) (cid:15) x / / ˜ F ( c ) We shall prove that the epimorphic family { e ( a,z ) : E ( a,z ) → E | ( a, z ) ∈ A c } satisfies the condition that for any ( a, z ) ∈ A c , the composite arrow x ◦ e ( a,z ) factors through χ Fc : F ( c ) → ˜ F ( c ) ; this will imply our thesis by Lemma6.39. As if E ( a,z ) ∼ = 0 then x ◦ e ( a,z ) factors through χ Fc : F ( c ) → ˜ F ( c ) ,we can suppose E ( a,z ) ≇ . Under this hypothesis, the arrow z : c → a ismonic in C . Indeed, for any two arrows f, g : b → c such that z ◦ f = z ◦ g , ˜ F ( f ) ◦ x ◦ e ( a,z ) = ˜ F ( f ) ◦ κ ( a,z ) ◦ h ( a,z ) = ˜ F ( f ) ◦ ˜ F ( z ) ◦ χ Fa ◦ h ( a,z ) = ˜ F ( g ) ◦ ˜ F ( z ) ◦ χ Fa ◦ h ( a,z ) = ˜ F ( g ) ◦ κ ( a,z ) ◦ h ( a,z ) = ˜ F ( g ) ◦ x ◦ e ( a,z ) , whence either f = g or the the equalizer of ˜ F ( f ) ◦ x and ˜ F ( g ) ◦ x is zero. But if the equalizerof ˜ F ( f ) ◦ x and ˜ F ( g ) ◦ x were zero then E ( a,z ) would also be isomorphic tozero (since it would admit an arrow to zero), contrary to our hypothesis; so f = g , as required. (cid:3) emark 6.44. By Proposition 6.35, Proposition 6.43 can be reformulatedas follows: for any object c ∈ C , F ( c ) ∼ = Hom E T C m -mod ( E ) ( γ ∗E ◦ y C c, ˜ F ) , where T C m is the geometric theory of flat functors on C op m . Theorem 6.45.
Let C be a small category and Flat m ( C op , E ) the subcategoryof Flat ( C op , E ) whose objects are the same as Flat ( C op , E ) and whose arrowsare the natural transformations between them which are pointwise monic.Then the extension functor ξ E : Flat ( C m op , E ) → Flat ( C op , E ) , along the embedding C m ֒ → C , which by Theorem 6.8 takes values into Flat m ( C op , E ) , is full on this latter category. Proof
Let
F, G be flat functors C m op → E with values in a Grothendiecktopos E , and let β : ˜ F → ˜ G be a pointwise monic natural transformationbetween them. We want to prove that there exists a natural transformation α : F → G such that β = ˜ α . It suffices to show that for any c ∈ C , β ( c ) :˜ F ( c ) → ˜ G ( c ) restricts (along the arrows χ Fc : F ( c ) → ˜ F ( c ) and χ Gc : G ( c ) → ˜ G ( c ) ) to an arrow F ( c ) → G ( c ) . To this end, we define a function γ E : Hom E ( E, F ( c )) → Hom E ( E, G ( c )) natural in E ∈ E . By Remark 3.2(iii),the set Hom E ( E, F ( c )) (resp. the set Hom E ( E, G ( c )) ) can be identifiedwith the set of arrows E → ˜ F ( c ) (resp. with the set of arrows E → ˜ G ( c ) )which factor through χ Fc (resp. through χ Gc ). By Proposition 6.43, for anyflat functor H : C op → E , the arrows E → ˜ H ( c ) which factor through χ Hc correspond precisely to the natural transformations γ ∗E /E ◦ y C c → ˜ H E whichare pointwise monic. Now, since β is pointwise monic then ! ∗ E ◦ β : ˜ F E → ˜ G E is also pointwise monic (since the functor ! ∗ E preserves monomorphisms, itbeing the inverse image of a geometric morphism); hence any generalizedelement E → ˜ F ( c ) which factors through χ Fc gives rise, by compositionwith ! ∗ E ◦ β of the corresponding, pointwise monic, natural transformation γ ∗E /E ◦ y C c → ˜ F E , to a poitwise monic natural transformation γ ∗E /E ◦ y C c → ˜ F E ,that is to a generalized element E → ˜ G ( c ) which factors through χ Gc . Butthis generalized element is precisely β c ◦ x . So β ( c ) : ˜ F ( c ) → ˜ G ( c ) restrictsto an arrow F ( c ) → G ( c ) , as required. (cid:3) We can now prove the following
Theorem 6.46.
Let T be a theory of presheaf type over a signature Σ . Thenthe injectivization of T satisfies condition ( iii )( b ) - (1) of Theorem 5.1 withrespect to the category f.p. T -mod ( Set ) m , and condition ( iii )( b ) - (2) with re-spect to the same category if every monic arrow in f.p. T -mod ( Set ) is sortwisemonic (for instance, by Proposition 6.47, if for every sort A over Σ thereexists the free T -model on A ). roof By Theorem 6.7, the composite functor ξ E : Flat ( f.p. T -mod ( Set ) m op , E ) → Flat ( f.p. T -mod ( Set ) op , E ) ≃ T -mod ( E ) , takes values in the subcategory T m -mod ( E ) of T -mod ( E ) . The functor ξ E is faithful by Proposition 3.2(iii) and full on T m -mod ( E ) by Theorem 6.45in view of the fact that for any sortwise monic T -model homomorphism f : M → N in E , the natural transformation α f : F M → F N correspondingto it under the Morita-equivalence Flat ( f.p. T -mod ( Set ) op , E ) ≃ T -mod ( E ) for T is pointwise monic. This latter fact can be proved as follows. Forany object D of the category f.p. T -mod ( Set ) , the value of α f at D can beidentified with the arrow [[ ~x . φ ]] M → [[ ~x . φ ]] N canonically induced by f , where φ ( ~x ) is ‘the’ formula which presents the model D ; therefore if f is sortwise monic then α f ( D ) is monic in E , it being the restriction of amonic arrow (namely f A × · · · × f A n , where ~x = ( x A , . . . , x A n ) ) along twosubobjects. (cid:3) The following proposition identifies a class of theories of presheaf type T with the property that the monic arrows of the category f.p. T -mod ( Set ) aresortwise monic. Proposition 6.47.
Let T be a theory of presheaf type over a signature Σ in which for every sort A over Σ the formula { x A . ⊤} presents a T -model,and let f : M → N be a homomorphism of finitely presentable T -models M and N . Then f is monic as an arrow of f.p. T -mod ( Set ) (equivalently, as anarrow of T -mod ( Set ) ) if and only if it is sortwise monic. Proof
Let A be a sort over Σ . As { x A . ⊤} presents a T -model F A thenfor any model P of T in Set we have an equivalence
Hom T -mod ( Set ) ( F A , P ) ∼ = P A natural in P . In particular we have equivalences Hom T -mod ( Set ) ( F A , M ) ∼ = M A and
Hom T -mod ( Set ) ( F A , N ) ∼ = N A under which the function f ◦ − : Hom T -mod ( Set ) ( F A , M ) → Hom T -mod ( Set ) ( F A , N ) corresponds to the func-tion f A : M A → N A . Now, if f is monic then the function f ◦ − : Hom T -mod ( Set ) ( F A , M ) → Hom T -mod ( Set ) ( P A , N ) is injective, equivalently f A : M A → N A is injective, as required. (cid:3)
In this section we shall establish a result providing a sufficient condition forthe injectivization of a theory of presheaf type of a certain form to be againof presheaf type. Before stating it, we need some preliminaries.The following definition gives a natural topos-theoretic generalization ofthe standard notion of congruence on a set-based structure.122 efinition 6.48.
Let Σ be a one-sorted first-order signature and M a Σ -structure in a Grothendieck topos E . An equivalence relation R M × M on M in E is said to be a congruence if for any function symbol f over Σ ofarity n , we have a commutative diagram R n (cid:15) (cid:15) / / ( M × M ) n ∼ = M n × M nf M × f M (cid:15) (cid:15) R / / M × M .
Proposition 6.49.
Let Σ be a one-sorted first-order signature and M a Σ -structure in a Grothendieck topos E . For any congruence R on M there existsa Σ -structure M/R on E whose underlying object is the quotient in E of M by the relation R , and a Σ -structure epimorphism p R : M → M/R givenby the canonical projection. Conversely, for any Σ -structure epimorphism q : M → N , the kernel pair R q of q is a congruence on M such that q isisomorphic to M/R q . Proof
The proof of the proposition is immediate by using the exactnessproperties of Grothendieck toposes relating epimorphisms and equivalencerelations. (cid:3)
Proposition 6.50.
Let T be a geometric theory over a one-sorted signature Σ and M a T -model in a Grothendieck topos E . If the axioms of T are all ofthe form ( φ ⊢ ~x ψ ) , where φ does not contain any conjunctions, then for anycongruence R on M the Σ -structure M/R is a T -model. Proof
The thesis can be easily proved by induction on the structure ofgeometric formulae over Σ , using the fact that the action of the canonicalprojection homomorphism p R : M → M/R on subobjects (of powers of M ) preserves the top subobject, the natural order on subobjects, imagefactorizations and arbitrary unions. (cid:3) The following lemma shows that one can always perform image factor-ization of homomorphisms of structures in regular categories.
Lemma 6.51.
Let Σ be a first-order signature and C a regular category.Then any Σ -structure homomorphism f : M → N in C can be factored as h ◦ g , where h : N ′ N is a Σ -substructure of N and g : M → N ′ issortwise a cover. Proof
First, we notice that in any regular category finite products of cov-ers are covers; indeed, composite of covers are covers (this can be easilyproved by using the definition of cover as an arrow orthogonal to the classof monomorphisms), and the product of two covers f × g , where f : A → B g : C → D , is equal to the composite (1 B × g ) ◦ ( f × C ) of two arrowswhich are pullbacks of covers.For every sort A over Σ we set N ′ A equal to Im ( f A ) , gA equal to thecanonical cover M A → Im ( f A ) and hA equal to the canonical subobject N ′ A N A . For any function symbol ξ : A , . . . A n → B over Σ , we set N ′ ξ equal to the restriction N ′ A ×· · ·× N ′ A n → N ′ B of N ξ : N A ×· · ·× N A n → N B . This restriction actually exists (and is unique) since f is a Σ -structurehomomorphism and gA × · · · × gA n is a cover. For any relation symbol R over Σ of type A , . . . , A n , we set N ′ R equal to the intersection of N R with the canonical subobject N ′ A × · · · × N ′ A n N A × · · · × N A n . It isclear that f = h ◦ g , that g is sortwise a cover and that h is a substructurehomomorphism, as required. (cid:3) The following proposition, giving an explicit characterization of decidableobjects in terms of their generalized elements, was stated in [26] as ExerciseVIII.8(a).
Proposition 6.52.
Let E be a cocomplete (in particular, a Grothendieck)topos and A an object of E . Then A is decidable if and only if for anygeneralized elements x, y : E → A there exists an epimorphic family (possiblyconsisting of just two elements) { e i : E i → E | i ∈ I } such that for any i ∈ I ,either x ◦ e i = y ◦ e i or the equalizer of x ◦ e i and y ◦ e i is zero. Proof
Let us suppose that A is decidable. Let p : P A × A be thecomplement of the diagonal subobject ∆ : A A × A . Consider the pullbackof h x, y i : E → A × A along p : E ′ s (cid:15) (cid:15) u / / P p (cid:15) (cid:15) E h x,y i / / A × A .
The equalizer i : R E ′ of x ◦ s and y ◦ s is zero; indeed, by definition of P , the diagram / / (cid:15) (cid:15) A ∆ (cid:15) (cid:15) P p / / A × A is a pullback, and the arrows z := x ◦ s ◦ i = y ◦ s ◦ i : R → A and u ◦ i : R → P satisfy the condition ∆ ◦ z = p ◦ u ◦ i .Let us denote by t : E ′′ E the pullback of the subobject ∆ : A A × A along h x, y i . The arrows s : E ′ → E and t : E ′′ → E are jointly epimorphic,they being the pullbacks of arrows which are jointly epimorphic, namely ∆ and p . 124herefore, by setting I = { , } , E = E ′ , E = E ′′ , e : E → E equal to s and e : E → E equal to t , we have that the family { e i : E i → E | i ∈ I } satisfies the condition in the statement of the proposition.Conversely, let us suppose that the condition in the statement of theproposition is satisfied. To prove that A is decidable, we have to show thatthe subobject ∆ ∨ ¬ ∆ A × A is an isomorphism, in other words that anysubobject h x, y i : E A × A factors through it.Notice that for any generalized elements x ′ , y ′ : E ′ → A , the arrow h x ′ , y ′ i : E ′ → A × A factors through ¬ ∆ A × A if and only if theequalizer of x ′ and y ′ is zero. Indeed, h x ′ , y ′ i factors through ¬ ∆ if andonly if its image does, and by definition of ¬ ∆ this holds if and only if thepullback of it (or equivalently, of h x ′ , y ′ i ) along ∆ is zero, i.e. if and only ifthe equalizer of x ′ and y ′ is zero.Now, by our hypotheses, there exists an epimorphic family { e i : E i → E | i ∈ I } such that for any i ∈ I , either x ◦ e i = y ◦ e i or the equalizer of x ◦ e i and y ◦ e i is zero. By Lemma 6.39, to prove that h x, y i factors through ∆ ∨ ¬ ∆ A × A it suffices to prove that for any i ∈ I , h x ◦ e i , y ◦ e i i factors through ∆ ∨ ¬ ∆ A × A . But by our hypothesis, for any given i ∈ I , either x ◦ e i = y ◦ e i (which implies that h x ◦ e i , y ◦ e i i factors through ∆ : A A × A ) or the equalizer of x ◦ e i and y ◦ e i is zero (which implies,as we have just seen, that h x ◦ e i , y ◦ e i i factors through ¬ ∆ A × A A );therefore for any i ∈ I , h x ◦ e i , y ◦ e i i factors through ∆ ∨ ¬ ∆ A × A , asrequired. (cid:3) Proposition 6.53.
Let Σ be a one-sorted signature, c a finite Σ -structurein Set , M a Σ -structure in a Grothendieck topos E whose underlying objectis decidable and E an object of E . Then for any Σ -structure homomorphism f : c → Hom E ( E, M ) there exists an epimorphic family { e i : E i → E | i ∈ I } in E and for each i ∈ I a quotient map q i : c → c i , where c i is a finite Σ -structure, and a disjunctive Σ -structure homomorphism (in the sense ofLemma 6.37) J i : c i Hom E ( E i , M ) such that J i ◦ q i = Hom E ( e i , M ) ◦ f for all i ∈ I : c f / / q i (cid:15) (cid:15) Hom E ( E, M ) f (cid:15) (cid:15) c i J i / / Hom E ( E i , M ) . Proof
Let us suppose that c has n elements x , . . . , x n . We know fromProposition 6.52 that for any pair ( i, j ) where i, j ∈ { , . . . , n } , there existarrows e ( i,j ) : E ( i,j ) → E and e ′ ( i,j ) : E ′ ( i,j ) → E such that e ( i,j ) and e ′ ( i,j ) arejointly epimorphic, f ( x i ) ◦ e ( i,j ) = f ( x j ) ◦ e ( i,j ) and f ( x i ) ◦ e ′ ( i,j ) , f ( x j ) ◦ e ′ ( i,j ) are disjoint. The iterated fibered product of all these epimorphic families(corresponding to the pairs ( i, j ) such that i, j ∈ { , . . . , n } ) thus yields an125pimorphic family { u k : U k → E | k ∈ K } such that for every k ∈ K thereexists a subset S k ⊆ { , . . . , n } × { , . . . , n } such that for every ( i, j ) ∈ S k , f ( x i ) ◦ u k = f ( x j ) ◦ u k and for every ( i, j ) / ∈ S k , f ( x i ) ◦ u k and f ( x j ) ◦ u k are disjoint. For each k ∈ K , consider the quotient q k : c → c q of c bythe congruence generated by the pairs of the form ( x i , x j ) for ( i, j ) ∈ S k .By definition of this congruence relation, the Σ -structure homomorphism Hom E ( e k , M ) ◦ f factors through q k , and the resulting factorization is adisjunctive Σ -structure homomorphism. We have thus found a set of datasatisfying the condition in the statement of the proposition, as required. (cid:3) Proposition 6.54.
Let T be a theory of presheaf type over a signature Σ such that the finitely presentable T -models coincide with the finitely pre-sentable T m -models. Suppose that the following condition is satisfied: forany Grothendieck topos E , object E of E and Σ -structure homomorphism x : c → Hom E ( E, M ) , where c is a finitely presentable T -model and M isa sortwise decidable T -model, there exists an epimorphic family { e i : E i → E | i ∈ I } in E and for each i ∈ I a T -model homomorphism f i : c → c i offinitely presentable T -models and a sortwise disjunctive Σ -structure homo-morphism x i : c i → Hom E ( E i , M ) such that x i ◦ f i = Hom E ( e i , M ) ◦ x forall i ∈ I . Then the injectification T m of T satisfies conditions ( i ) and ( ii ) ofTheorem 5.1 (with respect to its category of finitely presentable models). Proof
The proposition represents the particular case of Theorem 6.3 for thefaithful interpretation of T into its injectivization, with K = H equal to thecategory of finitely presentable T -models. The hypotheses of the theoremare satisfied since the functor R a is full (cf. Remark 6.4). (cid:3) Remark 6.55.
Any geometric theory T such that the finitely presentable T -models are precisely the finitely generated ones and all its axioms are of theform ( φ ⊢ ~x ψ ) , where ψ is a quantifier-free geometric formula, satisfies thefirst of the hypotheses of the Corollary. Indeed, every finitely presentable T -model is clearly finitely presentable as a T m -model, and since by Proposition6.11 the substructures of models of T are again models of T , every model of T can be expressed as the directed union of its finitely generated submodels;so the finitely presentable T m -models are precisely the finitely generated T -models. Corollary 6.56.
Let T be a theory of presheaf type over a signature Σ suchthat the finitely presentable T -models coincide with the finitely presentable T m -models and the monic T -model homomorphisms in Set are precisely thehomomorphisms which are sortwise monic. Suppose that the following condi-tion is satisfied: for any Grothendieck topos E , object E of E and Σ -structurehomomorphism x : c → Hom E ( E, M ) , where c is a finitely presentable T -model and M is a sortwise decidable T -model, there exists an epimorphic amily { e i : E i → E | i ∈ I } in E and for each i ∈ I a T -model ho-momorphism f i : c → c i of finitely presentable T -models and a sortwisedisjunctive Σ -structure homomorphism x i : c i → Hom E ( E i , M ) such that x i ◦ f i = Hom E ( e i , M ) ◦ x for all i ∈ I . Then the injectification T m of T isof presheaf type. Proof
By Theorem 6.46, T m satisfies condition ( iii ) of Theorem 5.1 (sinceall the the monic arrows in the category f.p. T -mod ( Set ) are injective func-tions), while Proposition 6.54 ensures that conditions ( i ) and ( ii ) of Theorem5.1 are satisfied. We can thus conclude that the theory T m is of presheaftype. (cid:3) As a consequence of Corollary 6.56, we obtain the following result.
Corollary 6.57.
Let T be a geometric theory over a one-sorted signature Σ such that any quotient of a finitely presentable T -model is a T -model (forinstance, a theory whose axioms are all of the form ( φ ⊢ ~x ψ ) , where φ does not contain any conjunctions - cf. Proposition 6.50). Suppose thatthe finitely presentable T -models are exactly the finite T -models and that allthe the monic arrows in the category f.p. T -mod ( Set ) are injective functions.Then if T is of presheaf type, T m is of presheaf type as well. Proof
Propositions 6.50 and 6.53 ensure that all the conditions of Corollary6.56 are satisfied. We can thus conclude that T m is of presheaf type, asrequired. (cid:3) In this section we introduce the syntactic notion of expansion of a geometrictheory, and show that it corresponds in a natural way to having a geometricmorphisms between the respective classifying toposes. Further, we charac-terize the expansions of theories whose corresponding geometric morphismsare localic and hyperconnected. This section has been inspired by section6.2 of [17], which contains a brief informal discussion of this topic.
Definition 7.1.
Let T be a geometric theory over a signature Σ .(i) A geometric expansion of T is a geometric theory obtained from T byadding sorts, relation or function symbols to the signature Σ and ge-ometric axioms over the resulting extended signature; equivalently, ageometric expansion of T is a geometric theory T ′ over a signature con-taining Σ such that every axiom of T , regarded as a geometric sequentin the signature of T ′ , is provable in T ′ .127ii) A geometric expansion T ′ of T is said to be localic if no new sorts areadded to Σ to obtain the signature of T ′ .(iii) A geometric expansion T ′ of T is said to be hyperconnected if no newfunction or relation symbols which only involve sorts over Σ are addedto Σ to form the signature of T ′ , and for any geometric sequent σ over Σ , σ is provable in T ′ if and only if it is provable in T .Notice that if T ′ is an expansion of a geometric theory T then there isa canonical morphism of sites ( C T , J T ) → ( C T ′ , J T ′ ) inducing a geometricmorphism p T ′ T : Set [ T ′ ] → Set [ T ] .We say that a geometric expansion T ′ of T is faithful (resp. fully faithful )if for every Grothendieck topos E , the induced functor ( p T ′ T ) E : T ′ -mod ( E ) → T -mod ( E ) is faithful (resp. full and faithful).We know from Theorem 9.1 [9] that the inclusion part of the surjection-inclusion factorization of the geometric morphism p T ′ T : Set [ T ′ ] → Set [ T ] canbe identified with the classifying topos of the quotient of T consisting of allthe sequents over Σ which are provable in the theory T ′ .Recall (cf. for instance section A4.6 of [21]) that a geometric morphism f : F → E is localic if every object of F is a subquotient (i.e. a quotientof a subobject) of an object of the form f ∗ ( a ) , where a is an object of E ;the morphism f is hyperconnected if f ∗ is full and faithful, and its image isclosed under subobjects in F .The following technical lemma will be useful in the sequel. Lemma 7.2.
Let f : F → E a geometric morphism between Grothendiecktoposes and C a full subcategory of E which is separating for E . Suppose thatthe following conditions are satisfied:(i) C is closed in E under finite products;(ii) f satisfies the property that the restriction f ∗ | C : C → F of f ∗ to C isfull and faithful;(iii) for any family of arrows T in C with common codomain, if the imageof T under f ∗ is epimorphic in F then T is epimorphic in E , and(iv) every subobject in F of an object of the form f ∗ ( c ) where c is an objectof C , is, up to isomorphism, of the form f ∗ ( m ) (where m is a subobjectin E ).Then f is hyperconnected. roof We have to prove that, under the specified hypotheses, f ∗ : E → F is full and faithful and its image is closed under subobjects in F .To prove that f ∗ is faithful, we have to verify that for any arrows h, k : u → v in E , f ∗ ( h ) = f ∗ ( k ) implies h = k . Since C is a separating set for E , wecan suppose without loss of generality u to lie in C . Now, h = k if and only ifthe equalizer z : w u of h and k is an isomorphism. The family of arrowsfrom objects of C to w is epimorphic, whence, as f ∗ ( z ) is an isomorphism(since f ∗ ( h ) = f ∗ ( k ) ), the family formed by the composition of these arrowswith z is epimorphic on u ; indeed, the members of this latter family all liein C and the image of this family under f ∗ is epimorphic. It follows that z is an epimorphism, equivalently an isomorphism, that is h = k , as required.To prove the fullness of f ∗ , we have to verify that for any objects a and b of E and any arrow s : f ∗ ( a ) → f ∗ ( b ) in F , there exists a (unique) arrow t : a → b in E such that f ∗ ( t ) = s . As C is a separating set for E there existepimorphic families { f i : c i → b | i ∈ I } and { g j : d j → a | j ∈ J } in E consisting of arrows whose domains lie in C .For any i ∈ I and j ∈ J , consider the pullback square r i,jq j (cid:15) (cid:15) p i / / f ∗ ( c i ) f ∗ ( f i ) (cid:15) (cid:15) f ∗ ( d j ) s ◦ f ∗ ( g j ) / / f ∗ ( b ) . By the universal property of pullbacks, the arrow h p i , q j i : r i,j → f ∗ ( c i ) × f ∗ ( d j ) ∼ = f ∗ ( c i × d j ) is a monomorphism. Since c i × d j lies in C by ourhypothesis, r i,j lies in the image of f ∗ and hence can be covered by arrows h i,jk : f ∗ ( e i,jk ) → r i,j (for k ∈ K i,j ) where each object e i,jk lies in C . Considerthe arrows p i ◦ h i,jk : f ∗ ( e i,jk ) → f ∗ ( c i ) and q j ◦ h i,jk : f ∗ ( e i,jk ) → f ∗ ( d j ) . Sincethe family { f ∗ ( g j ◦ m i,jk ) | i ∈ I, j ∈ J, k ∈ K i,j } is epimorphic in F , ourhypotheses ensure that the family { g j ◦ m i,jk : e i,jk → A | i ∈ I, j ∈ J, k ∈ K i,j } is epimorphic in E . To define an arrow t : a → b in E it is therefore equivalentto specify, for each i ∈ I , j ∈ J and k ∈ K i,j , an arrow v i,jk : e i,jk → b in sucha way that the compatibility conditions of Corollary 2.8 are satisfied. Take v i,jk equal to f i ◦ l i,jk . The compatibility conditions hold since the images ofthem under the functor f ∗ do and, as we have proved above, f ∗ is faithful.Thus we have a unique arrow t : a → b in E such that t ◦ g j ◦ m i,jk = f i ◦ l i,jk .The fact that f ∗ ( t ) = s follows at once.To conclude the proof of the lemma, it remains to show that the image of f ∗ is closed under subobjects in F . Let a be an object of E and k : z f ∗ ( a ) a subobject in F . Since C is a separating set for E , the canonical arrow fromthe coproduct of all objects in C to a is an epimorphism y ; by pulling back f ∗ ( y ) along the monomorphism k we obtain a monomorphism u from z ′ to dom ( f ∗ ( y )) and an epimorphism e : z ′ → z . Now, by our hypotheses, the129ullbacks of u along the coproduct arrows belong to the image of f ∗ ; fromthe fact that f ∗ preserves coproducts it thus follows that u itself belongsto the image of f ∗ , it being of the form f ∗ ( m ) , where m is a coproduct in E of subobjects in C . Now, since e is an epimorphism, z is isomorphic tothe coequalizer of its kernel pair r : w f ∗ ( b ) × f ∗ ( b ) ∼ = f ∗ ( b × b ) , where b = dom ( m ) . Using the coproduct representation of m and the fact that C isclosed under finite products in E , one can prove by considering the pullbacksof r along the images under f ∗ of the coproduct arrows to b × b , that r lies, upto isomorphism, in the image of f ∗ ; therefore, as f ∗ preserves coequalizers, k is isomorphic to a subobject in the image of f ∗ , as required. (cid:3) Theorem 7.3.
Let T be a geometric theory over a signature Σ and T ′ ageometric expansion of T over a signature Σ ′ . If T ′ is a hyperconnected (resp.a localic) expansion of T then p T ′ T : Set [ T ′ ] → Set [ T ] is a hyperconnected(resp. a localic) geometric morphism.In particular, the hyperconnected-localic factorization of the geometricmorphism p T ′ T : Set [ T ′ ] → Set [ T ] is given p T ′′ T ◦ p T ′ T ′′ , where T ′′ is the inter-mediate expansion of T obtained by adding to the signature Σ of T no newsorts and all the relation and function symbols over Σ ′ which only involvethe sorts over Σ , and all the sequents over this extended signature which areprovable in T ′ . Proof
Suppose that T ′ is localic over T . We have to prove that every objectof Set [ T ′ ] is a quotient of a subobject of an object of the form f ∗ ( H ) where H is an object of Set [ T ] . Since the signature of T ′ does not contain anysorts already present in the signature of T and the objects of the category C T ′ form a separating set for the topos Set [ T ′ ] , we can conclude that thesubobjects of the objects of the form p T ′ T ∗ ( { ~x . ⊤} ) form a separating setfor Set [ T ′ ] . Therefore any object of Set [ T ′ ] is a quotient of a coproduct ofsubobjects of objects of the form p T ′ T ∗ ( { ~x . ⊤} ) ∼ = { ~x . ⊤} ; but the fact that p T ′ T ∗ preserves coproducts immediately implies that any such coproduct is asubobject of an object in the image of p T ′ T ∗ .Suppose instead that T ′ is hyperconnected over T . We have to provethat p T ′ T ∗ is hyperconnected. It suffices to apply Lemma 7.2 by taking E equal to Set [ T ] and C equal to the syntactic category C T ; the fact that thehypotheses of the lemma are satisfied follows immediately from the fact that T ′ is hyperconnected over T . (cid:3) Theorem 7.4.
Let T be a geometric theory, E a Grothendieck topos and M a model of T in E . Then E is the classifying topos for T and M is a universalmodel for T if and only if the following conditions are satisfied:(i) The family F of objects which can be built from the interpretations in M of the sorts, function symbols and relation symbols over the signature f T by using geometric logic constructions (i.e. the objects given bythe interpretations in M of geometric formulae over the signature of T ) is separating for E ;(ii) The model M is conservative for T ; that is, for any geometric sequent σ over the signature of T , σ is valid in M if and only if it is provablein T ;(iii) Any arrow k in E between objects A and B in the family F of point ( i ) is definable; that is, if A (resp. B ) is equal to the interpretation ofa geometric formula φ ( ~x ) (resp. ψ ( ~y ) ) over the signature of T , thereexists a T -provably functional formula θ from φ ( ~x ) to ψ ( ~x ) such thatthe interpretation of θ in M is equal to k . Proof
By the universal property of the geometric syntactic category C T of T , the T -model M corresponds to a geometric functor F M : C T → E assigningto each object (or arrow) of C T its interpretation in M . Condition ( ii ) inthe statement of the theorem is equivalent to the assertion that the functor F M is faithful, while condition ( iii ) is equivalent to saying that F M is full.Therefore under conditions ( ii ) and ( iii ) , C T embeds as a full subcategory of E . Now, condition ( i ) ensures that C T is dense in E , whence the ComparisonLemma yields an equivalence E ≃ Sh ( C T , J ) , where J is the Grothendiecktopology on C T induced by the canonical topology on E , that is the geometrictopology on C T . By the syntactic construction of classifying toposes anduniversal models, we can thus conclude that E is ‘the’ classifying topos for T and M is a universal model for T . (cid:3) As immediate corollaries of Theorem 7.4, one recovers the followingknown results:(i) Let C be a separating set of objects for a Grothendieck topos E , and Σ C the signature consisting of one sort p c q for each object c of C andone function symbol p f q for each arrow f in E between objects in C .Then there exists a geometric theory T over the signature Σ C classifiedby E , whose universal model is given by the ‘tautological’ Σ C -structurein E (cf. p. 837 [21]);(ii) Let B be a pre-bound for E over Set (that is, an object such that thesubobjects of its finite powers form a generating set for E ); then thereexists a one-sorted geometric theory T classified by E and a universalmodel for T whose underlying object is B (cf. Theorem D3.2.5 [21]).The first result can be obtained from Theorem 7.4 by observing that thetheory T of the tautological Σ C -structure obviously satisfies all the hypothe-ses of the theorem. That the first two conditions are satisfied is obvious,131hile the fact that the third holds can be proved as follows. Since everyobject of the syntactic category of T is a subobject of a finite product ofobjects of the form { x p c q . ⊤} (for c ∈ C ), it is enough to prove that everyarrow k : R → c in E having as codomain an object c in C is definable. Sup-pose that R = [[ ~x . φ ]] M , where the sorts of the variables in ~x = ( x , . . . , x n ) are respectively p c ′ q , . . . , p c ′ n q , and denote by r : R → c ′ × · · · × c ′ n thecorresponding subobject. Since C is a separating set for E , the family ofarrows { f i : c i → R | i ∈ I } from objects of C to R is epimorphic; hence thegeometric formula _ i ∈ I ( ∃ z p c i q i )( x = p g q ( z i ) ∧ · · · ∧ x n = p g n q ( z i ) ∧ x p c q = p k ◦ f i q ( z i )) , where r ◦ f i = h g , . . . , g n i , is T -provably functional from { ~x . φ } to { x p c q . ⊤} and its interpretation coincides with k . By Diaconescu’s theorem, thetheory T can be explicitly characterized as the theory of flat J CE -continuousfunctors on C , where J CE is the Grothendieck topology on C induced by thecanonical topology on E .The second result can be deduced from Theorem 7.4 by taking T to bethe theory of the tautological structure over the one-sorted signature Σ B consisting of an n -ary relation symbol p R q for each subobject R B n in E .The fact that T satisfies the first two conditions of the theorem is obvious,while the validity of the third condition follows from the fact that the graphsof morphisms B n → B in E are interpretations of ( n + 1) -relation symbolsover Σ B .The following theorem provides a converse to Theorem 7.3. Theorem 7.5.
Let p : E →
Set [ T ] be a geometric morphism to the classi-fying topos of a geometric theory T . Then p is, up to isomorphism, of theform p T ′ T for some geometric expansion T ′ of T . If p is hyperconnected (resp.localic) then we can take T ′ to be a hyperconnected (resp. localic) expansionof T . Proof
Choose a triplet T = ( C ob , C arr , C rel ) consisting of a set C ob of objectsof E , of a set C arr of arrows in E from finite products of objects of C ob toobjects of C ob and of a set C rel of subobjects of finite products of objects of C ob with the property that the family of objects of E which can be built outof objects in C ob , arrows in C arr and subobjects in C rel by using geometriclogic constructions is separating for E . By definition of Grothendieck topos,such a triplet always exists. Let us define an expansion T T of T as follows:the signature Σ T T of T T is obtained by adding to the signature of T onesort p c q for each object c in C ob which is not of the form f ∗ ( H ) for someobject H of C T ֒ → Set [ T ] , one function symbol p f q for each arrow f in C arr whose domain or codomain is not of the form f ∗ ( H ) (with the obvious sorts),one relation symbol for each subobject in C rel (with the obvious sorts, the132nes corresponding to an object of the form f ∗ ( { ~x . φ } ) being the sorts ofthe variables ~x ) and an additional relation symbol p R q for any subobject R c × · · · × c n in E (where c , . . . , c n are objects in C ob ) which cannotbe obtained from the data in T by means of geometric logic constructions(whose sorts are the obvious ones).Consider the tautological T T -structure M in E obtained by interpretingeach sort p c q by the corresponding object c (and similarly for the functionand relation symbols added to the signature of T ), and each sort A over thesignature of T by the object f ∗ ( { x A . ⊤} ) (and similarly for the functionand relation symbols over the signature of T ). Define T ′ to be the theory of M over the signature T T . The theory T ′ satisfies the conditions of Theorem7.4; the validity of the first two conditions is obvious, while the validity ofthe third follows from the fact that any subobject of a product of objectsin C ob is definable in T ′ and the model M is conservative for T ′ , whencethe formula defining the graph of the given arrow is T ′ -provably functionalfrom the formula defining the domain to the formula defining the codomain.Therefore T ′ is classified by the topos E with universal model M . Noticethat T ′ is an expansion of T . This proves the first part of the theorem.Suppose now that f is localic. We can define a triplet T = ( C ob , C arr , C rel ) satisfying the conditions specified above as follows: we set C ob equal to theset of objects of the form f ∗ ( H ) where H is an object of C T ֒ → Set [ T ] , C arr equal to the empty set and C rel equal to the set of all subobjects of (finiteproducts of) objects in C ob . Alternatively, one can take C ob to be the set ofobjects of the form f ∗ ( { x A . ⊤} ) (where A is any sort over the signature of T ), C arr to be the set of arrows of the form f ∗ ([ θ ]) , where [ θ ] is an arrow in C T , and C rel to be the set of all subobjects of (finite products of) objects in C ob . In either case, the signature of the resulting expansion will contain nonew sorts with respect to the signature of T and hence it will be localic.Suppose instead that f is hyperconnected. Given any set of objects K of E which, together with the objects of the form f ∗ ( H ) (where H is anobject of C T ֒ → Set [ T ] ), form a separating set of E , we can define a triplet T = ( C ob , C arr , C rel ) satisfying the required conditions by setting C ob = K , C arr = ∅ and C rel = ∅ . Since f is hyperconnected, f ∗ is full and the imageof f ∗ is closed under subobjects; hence the signature of T T does not containany relation or function symbol only involving the sorts of T . On the otherhand, any geometric sequent over the signature of T is provable in T ′ if andonly if it is provable in f ∗ ( M T ) , where M T is the universal model of T lyingin Set [ T ] (since f ∗ is full and faithful), i.e. if and only if it is provable in T ′ .Hence T ′ is a hyperconnected expansion of T , as required.This completes our proof. (cid:3) Theorem 7.5 yields, in view of the equivalence between conditions ( iii )( b ) and ( iii )( c ) of Theorem 5.1 and Remark 5.2(d), the following reformulationof a particular case of Theorem 6.46.133 heorem 7.6. Let T be a theory of presheaf type such that the finitely pre-sentable models of T coincide with the finitely presentable models of T m .Then there is a faithful expansion (in the sense of section 7.1) of the in-jectivization of T which is classified by the topos [ f.p. T -mod ( Set ) m , Set ] .If moreover all the monic homomorphisms in T -mod ( Set ) are all sortwisemonic then this expansion can be taken to be fully faithful. (cid:3) A simple example of a theory satisfying both of the hypotheses of thetheorem is the theory A of commutative rings with unit. Indeed, the finitelypresented commutative rings with unit coincide precisely with the finitelygenerated ones, that is with the finitely presentable models of A m ; also, themonic ring homomorphisms are precisely the injective ones.Another consequence of Theorem 7.4 is the following criterion for a geo-metric theory to be of presheaf type. Theorem 7.7.
Let T be a geometric theory over a signature Σ . Then T isof presheaf type if and only if the following conditions are satisfied:(i) Every finitely presentable model is presented by a geometric formulaover Σ ;(ii) Every property of finite tuples of elements of a (finitely presentable) T -model which is preserved by T -model homomorphisms is definable bya geometric formula over Σ ;(iii) The finitely presentable T -models are jointly conservative for T . Proof
The fact that any theory of presheaf type satisfies the given conditionswas established in [13]. It thus remains to prove the ‘if’ part of the theorem.Consider the Σ -structure U in the topos [ f.p. T -mod ( Set ) , Set ] given bythe forgetful functors at each sort. Clearly, U is a T -model.To deduce our thesis, we shall verify that T satisfies the conditions ofTheorem 7.4 with respect to the model U .Since every finitely presentable T -model is presented by a geometric for-mula over Σ , the first condition of the theorem is satisfied; indeed, anyrepresentable functor Hom f.p. T -mod ( Set ) ( c, − ) is isomorphic to the interpre-tation of a formula φ ( ~x ) in the model U (take as φ ( ~x ) any formula presenting c ). The second condition of the theorem follows immediately from the factthat the finitely presentable T -models are jointly conservative for T . It re-mains to show that the third condition of the theorem is satisfied. To thisend, we observe that for any geometric formulae φ ( ~x ) and ψ ( ~y ) over thesignature of T , the graph of any arrow [[ ~x . φ ]] U → [[ ~y . ψ ]] U in the topos [ f.p. T -mod ( Set ) , Set ] is a subobject of the product [[ ~x . φ ]] U × [[ ~y . ψ ]] U in [ f.p. T -mod ( Set ) , Set ] and hence it defines a property of tuples of elements134f finitely presentable models of T which is preserved by T -model homomor-phisms; therefore, by our assumptions, there exists a formula θ ( ~x, ~y ) over Σ such that its interpretation in U coincides with such subobject. Since U is conservative and such subobject is the graph of an arrow in the topos [ f.p. T -mod ( Set ) , Set ] , the formula θ ( ~x, ~y ) is T -provably functional from φ ( ~x ) to ψ ( ~y ) , as required. (cid:3) In this section we shall discuss the problem of expanding a geometric theory T to a theory of presheaf type classified by the topos [ f.p. T -mod ( Set ) , Set ] .We shall say that a geometric theory is a presheaf completion of a geomet-ric theory T if it is an expansion of T such that the geometric morphism p T ′ T : Set [ T ′ ] → Set [ T ] is isomorphic to the canonical geometric morphism [ f.p. T -mod ( Set ) , Set ] → Set [ T ] .The results of section 7.1 show that in order to obtain a presheaf com-pletion of a given geometric theory T we can add a new sort p c q for eachfinitely presentable T -model c which is not presented by a geometric for-mula over the signature of T , and a relation symbol for each subobjectof a finite product of objects of [ f.p. T -mod ( Set ) , Set ] which are either ofthe form Hom f.p. T -mod ( Set ) ( c, − ) (where c is not presented by any geomet-ric formula over the signature of T ), or of the form U A (evaluation functorf.p. T -mod ( Set ) → Set at the sort A ) where A is a sort over the signatureof T (the sort corresponding to such a representable Hom ( c, − ) being p c q and to a functor U A being A ). Indeed, by Theorem 7.4, the theory of thetautological structure over this extended signature will be an expansion ofthe theory T classified by the topos [ f.p. T -mod ( Set ) , Set ] . Notice that if T satisfies the property that every finitely presentable model of T is presentedby a geometric formula over its signature then this expansion of T is localicover T .Of course, as it is clear from the results of section 7.1, there are in generalmany syntactic ways of ‘completing’ a given geometric theory to a theoryof presheaf type; the procedure described above represents just a particularchoice which is by no means canonical. In fact, what is most interesting inpractice is to obtain explicit axiomatizations of presheaf-type completions ofa given theory T directly from the axioms of T (cf. section 8 below for someexamples). Nonetheless, the results established above provide a useful guidein seeking such axiomatizations, as they indicate that in order to complete ageometric theory T to a theory classified by the topos [ f.p. T -mod ( Set ) , Set ] one should expand the language of T in order to make each finitely pre-sentable T -model presented by a formula in the extended signature and ev-ery property of finite tuples of elements of finitely presentable models of T definable by a geometric formula in the extended signature; we shall see con-135rete applications of these general remarks in section 8. In fact, Theorem 7.7ensures that making every finitely presentable model T finitely presented bya formula written in a possibly expanded signature is a necessary conditionfor the resulting theory to be classified by the topos [ f.p. T -mod ( Set ) , Set ] .Moreover, if one is able to prove that any finitely presentable T -model isstrongly finitely presented as a model of the theory T , when the latter isconsidered over an extended signature Σ ′ , condition ( iii ) of Theorem 5.1 isautomatically satisfied, while conditions ( i ) and ( ii ) can be made to holdat the cost of adding further axioms to T over the signature Σ ′ (cf. The-orem 6.29). On the other hand, by Theorem 7.7, in order to complete T to a theory of presheaf type classified by the topos [ f.p. T -mod ( Set ) , Set ] ,one should also expand the signature of T by adding relation symbols formaking any property of finitely presentable T -models which is preserved byhomomorphisms of models definable over the extended signature (if it is notalready definable over the original signature).The following lemma shows that, under appropriate conditions, modelswhich are finitely presented over a given signature remain finitely presentedover a larger signature obtained from the former by adding relation symbolscharacterized by disjunctive sequents of a certain form. Lemma 7.8.
Let T be a geometric theory over a signature Σ , Σ ′ a signatureobtained from Σ by only adding relation symbols R and S a geometric theoryover Σ ′ obtained from T by adding pairs of axioms of the form ( ⊤ ⊢ ~z R ( ~z ) ∨ W i ∈ I φ Ri ( ~z )) and ( R ∧ W i ∈ I φ Ri ( ~z ) ⊢ ~z ⊥ ) , where for each i ∈ I φ Ri is a geometricformula over Σ such that there exists a geometric formula ψ Ri ( ~z ) over Σ withthe property that the sequents ( φ Ri ∧ ψ Ri ⊢ ~z ⊥ ) and ( ⊤ ⊢ ~z φ Ri ∨ ψ Ri ) areprovable in S .Let R be the cartesianization of S (in the sense of Remark 6.23(d)) and φ ( ~x ) = φ ( x , . . . , x n ) a R -cartesian formula over Σ ′ with the property thatthere exists a R -model M φ with n generators ~ξ = ( ξ , . . . , ξ n ) ∈ [[ ~x . φ ]] M φ such that for any R -model N , the elements of the interpretation [[ ~x . φ ]] N are in natural bijective correspondence with the Σ -structure homomorphisms f : M φ → N such that f ( ~ξ ) ∈ [[ ~x . φ ]] N through the assignment f → f ( ~ξ ) .Then M φ is finitely presented by the formula { ~x . φ } as a R -model. Proof
We have to prove that, for any R -model N and any tuple ~a ∈ [[ ~x .φ ]] N , the unique Σ -structure homomorphism f : M φ → N such that f ( ~ξ ) = ~a preserves the satisfaction of all the relation symbols R in Σ ′ , i.e. that forany such symbol R of arity m and any m -tuple ( y , . . . , y m ) of elements of M φ satisfying R M φ , the m -tuple ( f ( y ) , . . . , f ( y m )) satisfies R N .As M φ is by our hypothesis generated by the elements ξ , . . . , ξ n , each y i is equal to the interpretation in M φ of a term t i ( ξ , . . . , ξ n ) evaluated in thetuple ~ξ = ( ξ , . . . , ξ n ) . For each i ∈ I , consider the formula ψ ′ iR ( x , . . . , x n ) obtained by replacing each of the variables z , . . . , z m in the formula ψ Ri with136he corresponding term t i . Let us show that the sequent φ ⊢ ~x ψ ′ Ri is valid inevery R -model. This amounts to showing that for any R -model P and anytuple ~b ∈ [[ ~x . φ ]] P , ~b satisfies the formula ψ ′ Ri . To prove this, we observe that,since M φ is by our hypotheses a S -model, the m -tuple ( y , . . . , y m ) satisfiesthe formula ψ Ri (for each i ∈ I ). Now, ( y , . . . , y m ) ∈ [[ ~y . ψ Ri ]] M φ means that ( ξ , . . . , ξ n ) ∈ [[ ~x . ψ ′ iR ]] M φ , which implies, since f ~b : M φ → P is a Σ -structurehomomorphism, that ~b ∈ [[ ψ ′ Ri ]] N , as required. Since R has enough modelsas it is cartesian, we can conclude that the sequent φ ⊢ ~x ψ ′ Ri is provablein R and hence in S (for each i ∈ I ). It follows that the cartesian sequent φ ⊢ ~x R ( t /z , . . . , t m /z m ) is provable in R . By evaluating this sequent in themodel N at the tuple f ( ~ξ ) ∈ [[ ~x . φ ]] N , and using the fact that Σ -structurehomomorphisms commute with the interpretation of Σ -terms, we obtain that ( f ( y ) , . . . , f ( y m )) = ( f ( t M φ ( ~ξ )) , . . . , f ( t M φ m ( ~ξ ))) = ( t N ( f ( ~ξ )) , . . . , t Nm ( f ( ~ξ ))) satisfies R N , as required. (cid:3) Notice that if a model of a geometric theory T is finitely presentable asa model of its cartesianization then it is strongly finitely presented. In fact,any structure of the form Hom E ( E, M ) , where M is a model of T in a topos E , is a model of the cartesianization of T , as it is obtained by applying aglobal section functor, which is cartesian, to a model of T . Conversely, ifa theory T is of presheaf type then any finitely presentable model of T isstrongly finitely presented (cf. Corollary 4.10) and hence finitely presentedrelatively to whatever sub-theory S of T (that is, theory S of which T is aquotient) whose set-based models admit representations as global sections Hom E ( E, M ) of models M of T in Grothendieck toposes (pairs of theoriessatisfying these conditions are investigated for instance in [14] and [16]).This remark can be often profitably applied to the cartesianization of T ,even though it is not known in general if it is always the case that everymodel of it admits a sheaf representation of the above kind.Summarizing, we have the following theorem. Theorem 7.9.
Let T be a theory of presheaf type and S be a sub-theory of T such that every set-based model of S admits a representation as a structureof global sections Hom E (1 , M ) of a model M of T in a Grothendieck topos E . Then every finitely presentable model of T is finitely presented as a modelof S . (cid:3) Let T be a geometric theory over a signature Σ and T ′ a geometric expansionof T over a signature Σ ′ . 137uppose that C is a subcategory of the category of finitely presentable T ′ -models such that the canonical functor ( p T ′ T ) Set : T ′ -mod ( Set ) → T -mod ( Set ) induced by the geometric morphism p T ′ T : Set [ T ′ ] → Set [ T ] restricts to afunctor j : C → f.p. T -mod ( Set ) . Then we have a commutative diagram Sh ( C T ′ , J T ′ ) p T ′ T / / Sh ( C T , J T )[ C , Set ] s T ′C O O [ j, Set ] / / [ f.p. T -mod ( Set ) , Set ] , t T O O where the geometric morphisms s T ′ C : [ C , Set ] → Sh ( C T ′ , J T ′ ) and t T : [ f.p. T -mod ( Set ) , Set ] → Sh ( C T , J T ) are the canonical ones and [ j, Set ] is the geometric morphism canonicallyinduced by the functor j . We shall refer to this diagram as to ( ∗ ) . Theorem 7.10.
Let T be a theory of presheaf type over a signature Σ and T ′ a geometric expansion of T over a signature Σ ′ . Suppose that T ′ is classifiedby the topos [ C , Set ] , where C is a full subcategory of f.p. T ′ -mod ( Set ) suchthat the functor ( p T ′ T ) Set : T ′ -mod ( Set ) → T -mod ( Set ) restricts to a faithfulfunctor j : C → f.p. T -mod ( Set ) . Then for any model c of T ′ in C whoseassociated T -model j ( c ) is finitely presented by a geometric formula φ ( ~x ) over Σ , there exists a geometric formula ψ ( ~x ) over Σ ′ in the context ~x whichpresents the T ′ -model c and such that the sequent ψ ⊢ ~x φ is provable in T ′ . Before giving the proof of the theorem, we need to recall the followingstraightforward lemma, of which we give a proof for the reader’s convenience.
Lemma 7.11.
Let R : A → B and L : B → A be a pair of adjoint functors,where R is the right adjoint and L is the left adjoint. Let η : 1 B → R ◦ L bethe unit of the adjunction and b an object of B . Then η ( b ) is monic if andonly if for any arrows f, g in B with codomain b , Lf = Lg implies f = g . Inparticular, η is pointwise monic if and only if the functor L is faithful. Proof
Let us denote by τ a,b the bijection between the sets Hom A ( Lb, a ) and Hom B ( b, Ra ) given by the adjunction. By the naturality in b of τ a,b , forany arrow h : b ′ → b in B , the arrow η b ◦ h corresponds under τ Lb,b ′ to thearrow Lh . Therefore, as τ Lb,b ′ is a bijection, η b ◦ f = η b ◦ g if and only if Lg = Lf . From this the thesis follows at once. (cid:3) Proof
The geometric morphism [ j, Set ] : [ C , Set ] → [ f.p. T -mod ( Set ) , Set ] is essential, that is its inverse image [ j, Set ] ∗ admits a left adjoint [ j, Set ] ! ,namely the left Kan extension along the functor j , and the following diagramcommutes: [ C , Set ] [ j, Set ] ! / / [ f.p. T -mod ( Set ) , Set ] C op y C O O j op / / f.p. T -mod ( Set ) op , y O O where y C and y are the Yoneda embeddings.The functor [ j, Set ] ! : [ C , Set ] → [ f.p. T -mod ( Set ) , Set ] satisfies the property that for any object c of C and any arrows α, β : P → y C ( c ) , where P is an object of [ C , Set ] , [ j, Set ] ! ( α ) = [ j, Set ] ! ( β ) implies α = β . Indeed, this is clearly true for P equal to a representable by thecommutativity of the above diagram, the full and faithfulness of the Yonedaembeddings y C and y ′ and the fact that the functor j is faithful by ourhypotheses, and one can always reduce to this case by considering a coveringof P in [ C , Set ] by representables.Now, by our hypotheses the geometric morphisms s T ′ C and t T definedabove are equivalences. Lemma 7.11 thus implies that the geometric mor-phism p T ′ T is essential and the unit of the adjunction between ( p T ′ T ) ∗ (rightadjoint) and ( p T ′ T ) ! (left adjoint) is monic when evaluated at any objectof the form y C T ′ ( { ~y . χ } ) , where χ ( ~y ) is a formula presenting a T ′ -model.Let c be a T ′ -model in C . Since T ′ is classified by the topos [ C , Set ] , c isfinitely presented by a formula { ~x . χ } over the signature Σ ′ of T ′ . Thecommutativity of the above square and of diagram ( ∗ ) thus implies thatthat ( p T ′ T ) ! ( y C T ′ ( { ~y . χ } )) ∼ = y C T ( { ~x . φ } ) , where φ ( ~x ) is a formula overthe signature Σ which presents the model j ( c ) and y C T : C T → Sh ( C T , J T ) , y C T ′ : C T ′ → Sh ( C T ′ , J T ′ ) are the Yoneda embeddings. By definition of thegeometric morphism p T ′ T , we have that ( p T ′ T ) ∗ ( y C T ( { ~x . φ } )) = y C T ′ ( { ~x . φ } ) (where the latter φ is considered as a formula over the signature Σ ′ ). Theunit of the adjunction between ( p T ′ T ) ∗ and ( p T ′ T ) ! thus yields a monic ar-row y C T ′ ( { ~y . χ } ) p T ′ T ∗ ( p T ′ T ! ( y C T ′ ( { ~y . χ } ))) ∼ = y C T ′ ( { ~x . φ } ) in the topos Sh ( C T ′ , J T ′ ) , in other words a monic arrow { ~y . χ } { ~x . φ } in the geomet-ric syntactic category C T ′ . Therefore { ~y . χ } is isomorphic, as an object of C T ′ , to an object { ~x . φ ′ } such that the sequent ( φ ′ ⊢ ~x φ ) is provable in T ′ ;hence φ ′ ( ~x ) presents c as a T ′ -model, as required. (cid:3) Examples
In this section we shall discuss in detail various non-trivial examples of the-ories of presheaf type in light of the theory developed in the paper.
As an application of Corollary 6.25, one can recover at once the well-knownresults that the following theories are of presheaf type:1. The theory of decidable objects (cf. [23] and p. 907 of [21]);2. The theory of decidable Boolean algebras (cf. Example D3.4.12 [21]);3. The theory of linear orders;4. The theory of total orders with endpoints (cf. section VIII.8 of [26])Notice that the former two theories are injectivizations of two cartesiantheories, namely the empty theory over a signature consisting exactly of onesort and the algebraic theory of Boolean algebras. The latter two theoriesclearly satisfy the hypothesis of Corollary 6.57; hence their injectivizationsare of presheaf type as well.
The theory C of abstract circles has been introduced by I. Moerdijk andshown in [28] to have the property that the points of Connes’ topos of cyclicsets (cf. [15]) can be identified with the set-based models of C . In thesame paper it is also stated that Connes’ topos actually classifies C , but theargument given therein seems incomplete.We shall prove in this section, by using Corollary 6.22, that C is ofpresheaf type classified by Connes’ topos. Specifically, we shall show that C satisfies the hypotheses of the corollary with respect to its Horn part (i.e.,the Horn theory consisting of the collection of all Horn sequents which areprovable in C ). We will also prove that the injectivization of C is of presheaftype as well.The signature Σ of the theory C consists of two sorts P and S (variablesof type P will be denoted by letters x, y, . . . , while variables of type S willbe denoted by letters a, b, . . . ), two unary function symbols P → S and P → S , two unary function symbols δ : S → P and δ : S → P , oneunary function symbol ∗ : S → S and a ternary predicate R of type S . Theaxioms of C can be formulated as follows:1. Non-triviality axioms : ( ⊤ ⊢ [] ( ∃ x )( x = x )); ⊤ ⊢ x,y ( ∃ a )( δ ( a ) = x ∧ δ ( a ) = y ));(0( x ) = 1( x ) ⊢ x ⊥ ) ‘Equational’ axioms : ( ⊤ ⊢ a a ∗∗ = a );( ⊤ ⊢ a δ ( a ∗ ) = δ ( a ));( ⊤ ⊢ x δ (0( x )) = x ∧ δ (0( x )) = x );( ⊤ ⊢ x x ) ∗ = 1( x ));( δ ( a ) = x ∧ δ ( a ) = x ⊢ x,a a = 0( x ) ∨ a = 1( x )); Axioms for concatenation : ( R ( a, b, c ) ∧ R ( a, b, c ′ ) ⊢ a,b,c,c ′ c = c ′ );( R ( a, b, c ) ⊢ a,b,c δ ( c ) = δ ( a ) ∧ δ ( c ) = δ ( b ));( R ( a, b, c ) ⊢ a,b,c R ( c ∗ , a, b ∗ )); R ( a, b, d ) ∧ R ( c, d, e ) ⊢ a,b,c,d,e ( ∃ e ′ )( R ( b, c, e ′ ) ∧ R ( a, e ′ , e ));( R ( a, b, x )) ⊢ a,b,x a = 0( x ));( δ ( a ) = x ⊢ a,x R (0( x ) , a, a );( δ ( a ) = δ ( b ) ⊢ a,b ( ∃ c ) R ( a, b, c ) ∨ ( ∃ d ) R ( b ∗ , a ∗ , d )) . A model of C in Set is said to be an abstract circle . Any set of points P of the circle S defines an abstract circle S P whose segments a ∈ S such that δ ( a ) = x and δ ( a ) = y are the oriented arcs on S from the point x to thepoint y . For any natural number n > , there is exactly one abstract circle,up to isomorphism, whose set of points has n elements; we shall denote it bythe symbol C n .To prove that C is of presheaf type, we first notice that the sequent ( δ ( a ) = δ ( b ) ∧ δ ( a ) = δ ( b ) ⊢ a,b a = b ) is provable in C . We shall refer to it as to the ‘uniqueness axiom’. Thissequent can be easily deduced as a consequence of the seventh axiom ofgroup and the fifth axiom of group .It follows that, modulo the non-triviality axioms and the uniqueness ax-iom, we can replace any expression of the form ( ∃ c ) φ ( c ) arising in an axiomof C with the requirement that the unique segment d such that δ ( d ) = δ ( c ) and δ ( d ) = δ ( c ) satisfies φ . In particular, the fourth axiom of group isprovably equivalent, modulo the non-triviality axioms and the uniquenessaxiom, to the following sequent: ( δ ( u ) = δ ( b ) ∧ δ ( u ) = δ ( c ) ∧ R ( a, b, d ) ∧ R ( d, c, e ) ⊢ a,b,c,d,e,u R ( b, c, u ) ∧ R ( a, u, e )) . is provably equivalent, modulothe non-triviality axioms and the uniqueness axiom, to the following sequent: ( δ ( c ) = δ ( a ) ∧ δ ( c ) = δ ( b ) ∧ δ ( a ) = δ ( b ) ⊢ a,b,c R ( a, b, c ) ∨ R ( b ∗ , a ∗ , c ∗ )) . From these remarks we see that C admits a presentation in which allthe axioms do not contain quantifications except for the first and second ofgroup .Let us show that C satisfies the hypotheses of Corollary 6.22 with respectto its Horn part and the category of finite C -models.Given a homomorphism f : c → Hom E ( E, M ) , where c is a finite modelof the Horn part of C , M is a model of C in a Grothendieck topos E and E isan object of E , we can ‘localize’ f (in the sense of section 6.4) a finite numberof times (once for the first axiom and once for each point of c for the secondaxiom) to obtain Σ -substructure homomorphisms f i : c i ֒ → Hom E ( E i , M ) such that the structures c i are finite models of the Horn part of C satisfyingthe non-triviality axioms (notice that any substructure of a structure ofthe form Hom E ( E i , M ) , where M is a model of C in E , satisfies all theHorn sequents provable in C ). We can clearly further localize each of thesehomomorphisms so to obtain the satisfaction of all the other axioms of C ;since this can be done without modifying their domains, the final result willbe a family of homomorphisms whose domains are structures which satisfyall the axioms of C .Next, we notice that for any set-based model M of the Horn part of C ,any point x ∈ M P and any segment a ∈ M S there exists a finite substructure N of M such that N P contains x and N S contains a (take N equal to thesubstructure of M given by the sets N P = { x, δ ( a ) , δ ( a ) } and N S = { a, x ) , x ) , δ ( a )) , δ ( a )) , δ ( a )) , δ ( a ))) } ) . Moreover, any two finite substructures N and N of M are contained in acommon substructure N of M (take N equal to the substructure of M givenby N P = N P ∪ N P and N S = N S ∪ N S ).This discussion, combined with the argument above (specialized to thecase E = Set ), shows that every model of C in Set is a directed union offinite models of C . It follows in particular that every finitely presentable C -model is finite (it being a retract of a finite model). On the other hand,every finite model of C is finitely presentable as a model of the Horn partof C (cf. Theorem 6.4 [6]). We can thus conclude that the hypotheses ofCorollary 6.22 are satisfied, whence C is of presheaf type classified by thetopos of covariant set-valued functors on the category of finite models of C .142ue to the presence of conjunctions in the premises of some axioms of C , we cannot directly apply Corollary 6.57 to conclude that the the injec-tivization C m of C is of presheaf type. We shall instead apply Corollary6.56. Since every C -model in Set is a directed union of finite C -models,the finitely presentable C m -models are exactly the finite C -models. More-over, by Proposition 6.47, the monic C -model homomorphisms in Set areprecisely the homomorphisms which are sortwise injective; indeed, the for-mulae { x P . ⊤} and { x S . ⊤} strongly present respectively the C -models C and C . To show that the hypotheses of Corollary 6.56 are satisfied, itremains to verify that for any Grothendieck topos E , object E of E and Σ -structure homomorphism x : c → Hom E ( E, M ) , where c is a finite C -modeland M is a sortwise decidable C -model, there exists an epimorphic family { e i : E i → E | i ∈ I } in E and for each i ∈ I a C -model homomorphism f i : c → c i of finite C -models and a sortwise disjunctive Σ -structure homo-morphism x i : c i → Hom E ( E i , M ) such that x i ◦ f i = Hom E ( e i , M ) ◦ x forall i ∈ I .In order to apply Proposition 6.53, we make C into a one-sorted theoryby identifying points x with the segments x and rewriting the axioms appro-priately. The proposition yields an epimorphic family { e i : E i → E | i ∈ I } in E and for each i ∈ I a sortwise surjective homomorphism q i : c → c i ,where c i is a finite Σ -structure, and a disjunctive Σ -structure homomor-phism (in the sense of Lemma 6.37) J i : c i Hom E ( E i , M ) such that J i ◦ q i = Hom E ( e i , M ) ◦ f for all i ∈ I . Now, since c is a C -model and q i is sortwise surjective, the structure c i satisfies the non-triviality axioms. Wecan clearly suppose E i ≇ without loss of generality, and hence the arrows J i to be injective. If we consider the image factorizations of the Σ -structure ho-momorphisms J i (in the sense of Lemma 6.51), we thus obtain substructures c ′ i Hom E ( E i , M ) whose underlying sets are the same as those of c i (sincethe J i are injective) and Σ -structure homomorphisms q ′ i : c → c ′ i . Since the c ′ i have the same underlying set as c i , they all satisfy the non-triviality ax-iom, and, at the cost of refining the epimorphic family { e i : E i → E | i ∈ I } ,we can suppose them to satisfy all the other axioms of C (cf. the argumentgiven above for showing that C satisfies the hypotheses of Corollary 6.22with respect to its Horn part and the category of finite C -models). So thehypotheses of Corollary 6.56 are satisfied, and we can conclude that C m is ofpresheaf type classified by the topos of covariant set-valued functors on thecategory of finite models of C and injective homomorphisms between them. In this section we shall revisit, from the point of view of the theory developedin the paper, a well-known example of a non-trivial theory of presheaf type,namely the geometric theory T of finite sets. Recall from [21] (ExampleD1.1.7(k)) that the signature Σ of T consists of one sort A and a n -ary143elation symbol R n for each n > . The axioms of T are the following: foreach n , one has the axiom σ n := ( R n ( x , . . . , x n ) ⊢ ~x,y _ ≤ i ≤ n y = x i ) , expressing the requirement that if an n -tuple of individuals satisfies the re-lation R n then it exhausts the members of (the set interpreting) the sort A .(The case n = 0 of this axiom is ( R ⊢ [] ⊥ ) , which says that if R holdsthen the interpretation of the sort A must be empty.) We also have, for each n > , the axiom ( ⊤ ⊢ [] _ ≤ i ≤ n ( ∃ x ) · · · ( ∃ x n ) R n ( x , . . . , x n )) . Finally, to ensure that the interpretations of the R n are uniquely determinedby that of the sort A (i.e. that R n holds for all n -tuples which exhaust theelements of the interpretation of sort A , and not just for some of them), oneadds the axioms ( R n ( x , . . . , x n )) ⊢ x ,...,x n R m ( x f (1) , . . . , x f ( m ) )) whenever f : { , , ..., m } → { , , ..., n } is a surjection, and ( R n ( x , . . . , x n ) ∧ x i = x j ⊢ x ,...,x n R n − ( x , . . . , x i − , x i +1 , . . . , x n )) whenever ≤ i < j ≤ n .We can deduce that T is of presheaf type as an application of Corollary6.22.The models of T in Set can be identified with the finite sets, while the T -model homomorphism are the precisely the surjective functions betweenthem.We can regard T as a quotient of its Horn part (i.e. of the theory consist-ing of all the Horn sequents over the signature of T which are provable in T ).The criterion for finite presentability given by Lemma 6.2 [6] ensures thatevery finite set, regarded as a model of T , is finitely presentable as a modelof the Horn part of T . Indeed, the finiteness of the structure immediatelyimplies that the second condition of the lemma is satisfied (cf. the proof ofTheorem 6.4 [6]), while the satisfaction of the first condition follows fromthe fact that every function from a finite model of cardinality n of the Hornpart of T to a set-based model M of the Horn part of T which preserves thepredicate R n preserves the predicate R m for any m < n (by the ‘introduc-tion’ and ‘elimination’ rules expressed by the last two groups of axioms for T ). To apply Corollary 6.22, it thus remains to prove that the second con-dition in the statement of the corollary is satisfied. Given a homomorphism144 : a → Hom E ( E, M ) , where a is a finite model of the Horn part of T and M is a T -model in a Grothendieck topos E , if the cardinality of aA is n thenfor any m > n , the sequent σ m is satisfied in a provided that σ n is. Indeed,if m ≥ n then for any m -tuple ( y , . . . , y m ) of elements in aA there exists asub- n -tuple of elements of aA obtained by removing m − n repetitions, whichsatisfies the relation R n if the m -tuple ( y , . . . , y m ) satisfies the relation R m ,since a , as a model of the Horn part of T , satisfies the Horn sequent express-ing the ‘elimination’ rule (i.e., the last group of axioms for T ). Similarly, byinvoking the ‘introduction’ rules, one can prove that if the cardinality of aA is n then for any k < n , the sequent σ k is satisfied in a provided that σ n is.Let us show that we can inductively localize the morphism f to eventuallyarrive at Σ -structure homomorphisms f i : a i → Hom E ( E i , M ) defined on Σ -structures a i which satisfy all the axioms of T . Notice that if the Σ -structurehomomorphisms h i : a → a i in such localization are quotient maps (in thesense that a tuple in a i satisfies a relation if and only if it is the image under h i of a tuple satisfying that relation in a ) then the fact that the a i satisfythe last two groups of axioms of T will follow automatically from the factthat a does. It will thus be enough to show, by the above considerations,that each a i satisfies the sequent σ n i , where n i is the cardinality of a i A , andthe second axiom of T .Starting from a Σ -structure homomorphism f : a → Hom E ( E, M ) , where a is a finite model of the Horn part of T , from the fact that M is a model of T it follows that there exists an epimorphic family { e i : E i → E | i ∈ I } andfor each i ∈ I a natural number n i and a n i -tuple ( ξ , . . . , ξ n i ) of generalizedelements E i → M A such that h ξ , . . . , ξ n i i factors through the interpretationof R n i in M . By taking a i to be the Σ -substructure of Hom E ( E i , M ) on thefinite subset consisting of the elements ξ , . . . , ξ n i plus all the elements in theimage of the homomorphism Hom E ( e i , M ) ◦ f , we clearly obtain a structuresatisfying the second axiom of T . This structure will also satisfy the lasttwo groups of axioms for T (the validity of Horn sequents is inherited bysubstructures). Now, in order to obtain from this family of Σ -structurehomomorphisms a localization such that the domains of its homomorphismssatisfy all the axioms of T , for each i ∈ I , it suffices to localize each f i a finitenumber of times (one for each ( n i + 1) -tuple ~u of elements of a i , where n i isthe cardinality of a i ), endowing the sets d i arising in the surjective-injectivefactorizations of the homomorphisms with the quotient structure inducedby the domains a i via the relevant quotient map q i (in the sense that theinterpretation of each relation symbol R over the signature of T in such a set d i is defined to be equal to the image of the interpretation of R in a i underthe quotient map q i ). By the above remarks, these structures will satisfy allthe axioms of T .This completes the proof of the fact that the hypotheses of Corollary 6.22are satisfied by the theory T with respect to its Horn part and the category C of finite T -models; therefore the theory T is of presheaf type classified by145he topos [ C , Set ] . As observed in [22], the coherent theory T of fields is not of presheaf type.In fact, one can easily identify two properties which are preserved by homo-morphisms of fields but which are not definable by geometric formulae inthe signature of the coherent theory of fields: the property of a field to havecharacteristic , and the property of a tuple ( x , . . . , x n , x n +1 ) of elements ofa field that the element x n +1 is transcendental over the subfield generatedby the elements x , . . . , x n . This can be easily seen by arguing as follows.Assuming the axiom of choice, the theory T has enough Set -based models (itbeing coherent). Hence if the property of having characteristic were defin-able by a geometric sentence φ over the signature of fields then the sequent ( ⊤ ⊢ [] φ ∨ W p ∈ P φ p ) , where P is the set of prime numbers and φ p (for each p ∈ P )is the sentence p . expressing the property of having characteristic p ,would be provable in T . But, T being coherent, the infinitary disjunctionon the right-hand side of the sequent would then be provably equivalent toa finite subdisjunction, which is absurd as it would imply that the set ofall possible characteristics of a field is finite. A similar argument works forthe other property, which, like the former, is the complement of a propertydefinable by a strictly infinitary geometric formula.In order to make such properties definable and possibly obtain a presheafcompletion of the theory T , it is thus necessary to enlarge the signature of T with new relation symbols. In fact, Johnstone introduces in [22] a -arypredicate R , expressing the property of a field to have characteristic , andfor each natural number n ≥ a n + 1 -predicate R n +1 ( x , . . . , x n , x n +1 ) expressing the property of x n +1 of being transcendental over the subfieldgenerated by the elements x , . . . , x n . Let Σ ′ be the resulting signature.Formally, one has to impose the following axioms over Σ ′ to ensure that thesepredicates have indeed the required meaning (below we use the abbreviation Inv ( z ) for the formula ( ∃ x )( x · z = 1 ∧ z · x = 1) ): ( ⊤ ⊢ R ∨ _ p ∈ P p . R ∧ p . ⊢ ⊥ ) (for each p ∈ P ); ( ⊤ ⊢ x ,...,x n +1 R n ( x , . . . , x n , x n +1 ) ∨ ( _ m ∈ N , ~c n ,..., ~c nm ( m X j =0 P ~c nj x jn +1 = 0) ∧ _ i ∈{ , ,...,m } Inv ( P ~c ni ))) (for each natural number n ≥ ), where the former disjunction is taken overall the natural numbers m ≥ and all the tuples ~c ni (for i ∈ { , . . . , m } ) of146nteger coefficients (i.e., coefficients of the form · · · + 1 an integer numberof times) of polynomials in n variables x , . . . , x n of degree ≤ m and theexpression P ~c ni (for each i ∈ { , . . . , m } ) denotes the polynomial term in thevariables x , . . . , x n corresponding to the tuple ~c ni , and ( R n ( x , . . . , x n , x n +1 ) ∧ ( _ m ∈ N , ~c n ,..., ~c nm ( m X j =0 P ~c nj x jn +1 = 0) ∧ _ i ∈{ , ,...,m } Inv ( P ~c ni ))) ⊢ x ,...,x n +1 ⊥ ) . Let D be the theory, called in [22] of Diers fields , obtained from T byadding these new predicates and the above-mentioned axioms.Note that the geometric formula _ m ∈ N , ~c n ,..., ~c nm ( m X j =0 P ~c nj x jn +1 = 0) ∧ _ i ∈{ , ,...,m } Inv ( P ~c ni )) is D -provably equivalent to a disjunction of geometric formulae, namely,the formulae ( m P j =0 P ~c nj x jn +1 = 0) ∧ Inv ( P ~c ni ) (for each m , ~c n , . . . , ~c nm and i ∈ { , . . . , m } ), each of which has a D -provable complement, namely theformula Inv ( m P j =0 P ~c nj x jn +1 = 0) ∨ P ~c ni = 0 .Notice also that, for any tuples ~c ni (for i ∈ { , . . . , m } ) of integer coeffi-cients of polynomial expressions P ~c ni in the variables x , . . . , x n , the cartesiansequent ( ∗ ) ( R n ( x , . . . , x n , x n +1 ) ∧ Inv ( P ~c ni ) ⊢ x ,...,x n +1 Inv ( m X j =0 P ~c nj x jn +1 = 0)) is provable in D .Following [22], we observe that the theory D satisfies the property thatevery finitely presentable model of it, i.e. any finitely generated field, isfinitely presented as a model of its cartesianization. This will follow fromLemma 7.8 once we have proved that every finitely generated field F ispresented by a finite set of generators, in the weak sense of the lemma, by aformula over the signature of D . To this end, we regard T as axiomatized inthe signature of von Neumann regular rings, which contains a unary functionfor the operation of pseudoinverse; indeed, over this signature, every finitelygenerated field (in the sense of field theory) becomes finitely generated (inthe sense of model theory).Notice that, by the above remarks, D satisfies the first set of hypothesesof Lemma 7.8 with respect to the theory T .We shall prove that every finitely generated field F is (weakly) presented(as a model of the cartesianization of D ) by a geometric formula over the147ignature of D by induction on the number n of generators of F . If n = 0 then F is equal to its prime field; so, either F has characteristic p , in whichcase it is equal to Z p , or F has characteristic , in which case it is equalto Q . Now, the field Z p is clearly (weakly) presented by the formula p . , while the field Q is (weakly) presented by the formula R since thesequent R ⊢ Inv ( n . is provable in the cartesianization of D for eachnon-zero natural number n . Now, consider a field F generated by n + 1 elements x , . . . , x n , x n +1 . If we denote by F its prime field, we have that F = F ( x , . . . , x n )( x n +1 ) .Suppose that F ( x , . . . , x n ) is (weakly) presented by a formula in n variables φ ( ~x ) with the elements x , . . . , x n as generators.There are two cases: either the element x n +1 is transcendental over thefield F ( x , . . . , x n ) or not.In the first case, F is isomorphic to the field of rational functions inone variable with coefficients in F ( x , . . . , x n ) , and it is (weakly) presentedby the formula φ ∧ R n ( x , . . . , x n , x n +1 ) with generators x , . . . , x n , x n +1 ;in other words, for any model ( A, { ( R n ) A n ∈ N } ) of the cartesianizationof D , the function which assigns a ring homomorphism f : F → A withthe property that f ( x , . . . , x n , x n +1 ) ∈ ( R n +1 ) A ∩ [[ ~x . φ ]] A to the element f ( x , . . . , x n , x n +1 ) is injective and surjective on ( R n +1 ) A ∩ [[ ~x . φ ]] A . Thiscan be proved as follows. Since F is generated by the elements x , . . . , x n +1 ,the injectivity is clear, so it remains to prove the surjectivity, i.e. that forany ( n + 1) -tuple ( a , . . . , a n , a n +1 ) ∈ ( R n +1 ) A ∩ [[ ~x . φ ]] A there exists a ringhomomorphism F → A which sends x i to a i for each i ∈ { , . . . , n + 1 } .By the induction hypothesis, since ( a , . . . , a n ) ∈ [[ ~x . φ ]] A , there exists aunique ring homomorphism g : F ( x , . . . , x n ) → A such that g ( x i ) = a i for each i ∈ { , . . . , n } . By definition of F , there exists a ring homo-morphism f : F → A which extends g and sends x n +1 to a n +1 if andonly if for every polynomial P with a non-zero coefficient P ~c ni ( x , . . . , x n ) in F ( x , . . . , x n ) , m P j =0 g ( P ~c nj ) a j is invertible in A . But since P ~c ni ( x , . . . , x n ) is non-zero (equivalently, invertible) in the field F ( x , . . . , x n ) , its image g ( P ~c ni ( x , . . . , x n )) = P ~c ni ( g ( x ) , . . . , g ( x n )) under the homomorphism g isinvertible in A ; therefore, since the sequent ( ∗ ) holds in A , the condition ( f ( x ) , . . . , f ( x n ) , a ) ∈ ( R n ) A entails the fact that m P j =0 g ( P ~c nj ) a j is invertiblein A , as required.In the second case, consider the minimal polynomial P for x n +1 over F ( x , . . . , x n ) ; then F is isomorphic to the quotient of F ( x , . . . , x n ) bythe ideal generated by the polynomial P . It is immediate to see that F is(weakly) presented by the formula in n + 1 variables φ ∧ P ( x n +1 ) = 0 withgenerators x , . . . , x n , x n +1 .These arguments show that the theories T and D satisfy the hypotheses148f Lemma 7.8. It follows that all the finitely generated fields are finitelypresented models of the cartesianization of D , as required. Condition ( iii ) of Theorem 5.1 is therefore satisfied (cf. Proposition 5.11(i)). Alternatively,we could have deduced the fact that the theory D satisfies condition ( iii ) ofTheorem 5.1 from Corollary 5.16 and Theorem 5.21.In [22], Johnstone shows that D is a theory of presheaf type by assuminga form of the axiom of choice to ensure that D has enough set-based mod-els. Theorem 5.1 allows to prove that D is of presheaf type directly, withoutassuming any non-constructive principles. Having already proved that con-dition ( iii ) of the theorem is satisfied, it remains to see that conditions ( i ) and ( ii ) hold as well.The fact that condition ( ii )( a ) of Theorem 5.7 holds follows immediatelyfrom the above-mentioned discussion in view of Remark 5.8(b). Condition ( ii )( c ) of Theorem 5.3 is automatically satisfied while condition ( ii )( a ) ofTheorem 5.7 follows from condition ( ii )( a ) of Theorem 5.7 (cf. Remarks5.4(b)-(c)), By Remark 5.8(a), to prove that condition ( ii )( b ) of Theorem5.7 holds, it suffices to verify that condition ( ii )( b ) of Theorem 5.3 does.It thus remains to verify that condition ( ii )( b ) of Theorem 5.3 holds, i.e.that for any finitely generated fields c and d , D -model M in a Grothendiecktopos E and Σ ′ -structure homomorphisms x : c → Hom E ( E, M ) and y : d → Hom E ( E, M ) , there exists an epimorphic family { e i : E i → E | i ∈ I } in E and for each i ∈ I a finitely generated field b i , field homomorphisms u i : c → b i , v i : d → b i and a Σ ′ -structure homomorphism z i : b i → Hom E ( E i , M ) such that Hom E ( e i , M ) ◦ x = z i ◦ u i and Hom E ( e i , M ) ◦ y = z i ◦ v i .We can prove this by induction on the sum n of the minimal number ofgenerators of c and of d .Before proceeding with the proof, it is convenient to remark the follow-ing fact: for any non-zero model ( A, { ( R n ) A n ∈ N } ) of the cartesianizationof D (for instance, a Σ ′ -structure of the form Hom E ( E, M ) , where M is amodel of D in the topos E and E is a non-zero object of E ) and any field e , considered as a model of D , all the Σ ′ -structure homomorphisms e → A reflect the satisfaction of the relations R n , i.e. f ( x , . . . , x n ) ∈ ( R n ) A implies ( x , . . . , x n ) ∈ ( R n ) e . This easily follows from the disjunctive axioms of D defining R n and the cartesian axiom ( ∗ ) . Note also that such homomor-phisms are always injective (since their domain is a field and their codomainis a non-zero ring).In proving our claim, we can suppose without loss of generality all theobjects E arising in Σ ′ -structure homomorphisms to structures of the form Hom E ( E, M ) to be non-zero (since removing zero arrows from an epimorphicfamily leaves the family epimorphic).If n = 0 (that is, if both c and d are equal to their prime fields) then c and d have the same characteristic; indeed, equalities of the form n . are preserved and reflected by the homomorphisms x and y (cf. the aboveremarks). So c and d are isomorphic, whence the claim is trivially satisfied.149et us now assume that the condition is true for all k ≤ n and prove it for n + 1 . We can represent c as c ′ ( x ) in such a way that a set of generatorsfor c can be obtained by adding x to a set of generators for c ′ ; then by theinduction hypothesis there exists an epimorphic family { e i : E i → E | i ∈ I } and for each i ∈ I a finitely generated field u i , field homomorphisms f i : c ′ → u i and g i : d → u i and a Σ ′ -structure homomorphism r i : u i → Hom E ( E i , M ) such that r i ◦ f i = Hom E ( E, M ) ◦ f | c ′ and r i ◦ g i = Hom E ( E, M ) ◦ g . Now,consider for each i ∈ I the element f ( x ) ◦ e i ∈ Hom E ( E i , M ) . Supposethat c ′ has m generators χ , . . . , χ m ; then for each i ∈ I , by the disjunctiveaxiom of D involving R m , there exists an epimorphic family { f i,j : F i,j → E i | j ∈ J i } such that for any j ∈ J i , either R m (( r i ◦ f i )( χ ) ◦ e i ◦ f i,j , . . . , ( r i ◦ f i )( χ m ) ◦ e i ◦ f i,j , f ( x ) ◦ e i ◦ f i,j ) or f ( x ) ◦ e i ◦ f i,j is the root of a non-zeropolynomial P with coefficients belonging to the von Neumann regular sub-ring of Hom E ( F i,j , M ) generated by the elements ( r i ◦ f i )( χ ) ◦ e i ◦ f i,j , . . . , ( r i ◦ f i )( χ m ) ◦ e i ◦ f i,j . In the first case, f ( x ) ◦ e i ◦ f i,j is transcendental over c ′ via the embedding Hom E ( f i,j , M ) ◦ r i ◦ f i and over u i via the embedding Hom E ( f i,j , M ) ◦ r i (apply the above remarks to these two embeddings); itfollows that the homomorphism f i extends to a homomorphism from c = c ′ ( x ) to u i .In the second case, the homomorphism Hom E ( e i ◦ f i,j , M ) ◦ f beinginjective, there exists a non-zero polynomial P with coefficients in c ′ suchthat f ( x ) ◦ e i ◦ f i,j is a root of the image of P under Hom E ( f i,j , M ) ◦ r i ◦ f i = Hom E ( e i ◦ f i,j , M ) ◦ f | c ′ . It follows that x is a root of P in c . Wecan thus clearly suppose P to be irreducible without loss of generality, andrepresent c = c ′ ( x ) as c = c ′ [ z ] /P ( z ) , via an isomorphism sending z to x ;denoting P ′ the image of P under the homomorphism f i , we thus obtainthat the arrow f i yields an arrow c = c ′ [ z ] /P ( z ) → u i [ w ] /P ′ ( w ) and thehomomorphism Hom E ( f i,j , M ) ◦ r i factors through the quotient map u i → u i [ w ] /P ′ ( w ) yielding a ring homomorphism which is in fact a Σ ′ -structurehomomorphism (by Lemma 7.8, cf. the argument given above for provingthat every finitely generated field is finitely presented as a model of thecartesianization of D ).From these remarks it is now straightforward to obtain a set of datasatisfying the requirements of our condition.We could have alternatively proved that D satisfies condition ( ii )( b ) ofTheorem 5.3 either by using Remark 5.4(d) or by applying Theorem 5.5. Let F be a field. We define the theory T F of algebraic extensions of F as theexpansion of the coherent theory of fields obtained by adding one constantsymbol a for each element a ∈ F and the following axioms: ( ⊤ ⊢ F = 1); ⊤ ⊢ F = 0);( ⊤ ⊢ a + b = a + F b ) , for any elements a, b ∈ F (the symbol + F denotes the addition operation in F ); ( ⊤ ⊢ a · b = a · F b ) , for any elements a, b ∈ F (the symbol · F denotes the multiplication operationin F ), plus the algebraicity axiom ( ⊤ ⊢ x _ n ∈ N ,a ,...,a n − ,a n ∈ F a n x n + a n − x n − + · · · + a = 0) . We shall prove that T F is of presheaf type. Clearly, the finitely pre-sentable models of T F are exactly the finitely generated algebraic extensionsof F , that is the finite extensions of F . One can prove, by adapting theargument used in the proof of the result that every finitely generated field isfinitely presented as a model of the cartesianization of the theory of Diers’fields established in section 8.2, that every finite extension of F is finitelypresented as a model of the cartesianization of T F . Specifically, every finiteextension F ( x , . . . , x n ) of F is presented by the conjunction of the formulaeof the form P i ( x , . . . , x i +1 )( x i ) = 0 (for i = 0 , . . . , n − ), where P is theminimal polynomial for the element x i +1 over the field F ( x , . . . , x i ) .Condition ( iii ) of Theorem 5.1 is thus satisfied by the theory T F withrespect to the category of finite extensions of F (cf. Proposition 5.11(i)).In verifying that conditions ( i ) and ( ii ) of Theorem 5.1 are satisfied, oneis reduced as in section 8.2 to check that condition ( ii )( b ) of Theorem 5.3holds; this can be done again by adapting the argument given in section 8.2to this case. We can thus conclude that T F is of presheaf type classified bythe topos of covariant set-valued functors on the category of finite extensionsof F .Next, let us consider the theory S F of T F of separable extensions of F ,that is the quotient of T F obtained by adding the following sequent: ( ⊤ ⊢ x _ n ∈ N , ( a ,...,a n − ,a n ) ∈S nF a n x n + a n − x n − + · · · + a = 0) , where S nF is the set of n -tuples of elements a , . . . , a n of F such that thepolynomial a n Z n + a n − Z n − + · · · , + a ∈ F [ Z ] is irreducible and separable.Clearly, the finitely presentable S F -models are precisely the finite sep-arable extensions of F . In particular, every finitely presentable S F -modelis finitely presentable as a T F -model. In fact, by Artin’s primitive elementtheorem, every finite separable extension of F is presented by a formula ofthe form { x . a n x n + a n − x n − + · · · + a = 0 } .The hypotheses of Corollary 6.22 are trivially satisfied; indeed, for any T F -model homomorphism M → N (in an arbitrary Grothendieck topos), if151 is separable (i.e., a model of S F ) then M is a fortiori separable as well.We can thus conclude that also the theory S F is of presheaf type, classifiedby the category of covariant set-valued functors on the category of finiteseparable extensions of F . Remark 8.1.
The theory of fields of a fixed finite characteristic p which arealgebraic over their prime field, which was proved in [11] to be of presheaftype classified by the category of covariant set-valued functors on the cate-gory of finite fields of characteristic p , is (trivially) Morita-equivalent to thetheory T Z p introduced above. In this section we shall study the injectivization G of the (algebraic) theoryof groups.Clearly, the finitely presentable G -models are precisely the finitely gen-erated groups.Even if G satisfies condition ( iii ) of Theorem 5.1 with respect to thecategory of finitely generated groups and injective homomorphisms betweenthem (by Theorem 5.21 and Corollary 5.16), G is not of presheaf type. To seethis, consider the property of an element x of a group G to be non-nilpotent.This property is clearly preserved by injective homomorphisms of groups, soif G were of presheaf it would be definable by a geometric formula φ ( x ) overthe signature of G . Then the sequent ( ⊤ ⊢ x _ n ∈ N φ ∨ ( x n = 1)) , would be provable in G (since G has enough set-based models and this se-quent is valid in every set-based G -model by definition of φ ) and hence, as G is coherent, the disjunction on the right hand side would be G -provablyequivalent to a finite sub-disjunction; but this is absurd since it implies thatthere exists a natural number n such that every nilpotent element x of agroup satisfies x n = 1 .To obtain a presheaf completion of the theory G , we add to the signatureof G a relation symbol R nN for each natural number n and any normal sub-group of the free group F n on n generators, and the following axioms (wherethe symbol = denotes the predicate of G which is G -provably complementedto the equality relation): ( ⊤ ⊢ ~x R nN ( ~x ) ∨ ( _ w ( ~x ) ,w ′ ( ~x ) ∈ F n | ww ′− ∈ N w = w ′ ) ∨ ( _ w ( ~x ) ,w ′ ( ~x ) ∈ F n | ww ′− / ∈ N w = w ′ )) and ( R nN ( ~x ) ∧ ( _ w ( ~x ) ,w ′ ( ~x ) ∈ F n | ww ′− ∈ N w = w ′ ∨ _ w ( ~x ) ,w ′ ( ~x ) ∈ F n | ww ′− / ∈ N w = w ′ ) ⊢ ~x ⊥ ) n ∈ N and any normal subgroup N of F n ), and ( ⊤ ⊢ ~x _ N ∈N n R nN ( ~x )) , where N n is the set of normal subgroups of F n (for any n ∈ N ).Let G p be the resulting theory; we shall prove that it is of presheaf type.Notice that the theories G and G p satisfy the first set of hypotheses ofLemma 7.8.Let us first show that every finitely generated group is finitely presentedas a model of the cartesianization of G p . By Lemma 7.8, to prove that F n /N is presented by the formula R nN ( ~x ) as a model of the cartesianization of G p ,it suffices to verify that for any set-based model ( G ′ , { ( R nN ) G | n ∈ N , N ∈N n } ) of the cartesianization of G p , denoting by ~ξ = ( ξ , . . . , ξ n ) the gen-erators of the group F n , the group homomorphisms f : F n /N → G ′ suchthat f ( ~ξ ) ∈ [[ ~x . R nN ]] G ′ correspond exactly to the n -tuples ~y of elementsof G ′ which belong to the interpretation in G ′ of the formula R nN (via theassignment f → f ( ~ξ ) ). Given a n -tuple ~y of elements of G ′ which belongto the interpretation of R nN in G ′ , we can define a function f : F n /N → G ′ by setting f ([ w ]) = w G ′ ( ~y ) . This is a well-defined injective group homomor-phism since the following sequents are provable in the cartesianization of G p and hence are valid in G ′ : ( R nN ⊢ ~x w = w ′ ) for any w, w ′ ∈ F n such that ww ′− ∈ N , and ( R nN ⊢ ~x w = w ′ ) for any w, w ′ ∈ F n such that ww ′− / ∈ N . Since every finitely generatedgroup is, up to isomorphism, of the form F n /N for some natural number n and a normal subgroup N of F n , we can conclude that the theory G p satisfiescondition ( iii ) of Theorem 5.1 (cf. Proposition 5.11(i)). An alternative wayto prove this would have been to invoke Corollary 5.16 and Theorem 5.21.As in the case of Diers fields treated in section 8.2, in order to verify thatthe theory G p satisfies conditions ( i ) and ( ii ) of Theorem 5.1, one is reducedto show that condition ( ii )( b ) of Theorem 5.3 is satisfied; but this followsfrom Remark 5.4(d). The theory G p is thus of presheaf type classified bythe topos [ f.g. Grp , Set ] , where f.g. Grp is the category of finitely generatedgroups and injective homomorphisms between them.The category f.g.
Grp is cocartesian; indeed, it has an initial object(namely, the trivial group) and pushouts (given by the free product withamalgamation construction, cf. [30]). It follows the topos [ f.g. Grp , Set ] iscoherent. The theory G p , in spite of being infinitary, is thus classified bya coherent topos. This fact has various implications for G p . For instance,153he coherence of the classifying topos for G p implies that every formula overthe signature of G p presenting a finitely generated group (regarded as a G p -model) is not only G p -irreducible, but also a coherent object of the classifyingtopos; in particular, for any G p -compact formula { ~x . φ } over the signatureof G p , the formula { x, y . φ ( x ) ∧ φ ( y ) } is also G p -compact (recall that aformula { ~z . χ } over the signature Σ of a geometric theory T is T -compact ifwhenever { ~z . χ } T -provably entails a disjunction of geometric formulas over Σ , { ~z . χ } T -provably entails a finite sub-disjunction of it). Semantically, aformula { ~x . φ } is G p -compact if whenever { S i | i ∈ I } is a family of assign-ments G → S Gi ⊆ G n sending each finitely generated group G to a subset S Gi ⊆ [[ ~x . φ ]] G in such a way that every injective homomorphism f : G → G ′ of groups sends tuples in S Gi to tuples in S G ′ i , if [[ ~x . φ ]] G = S i ∈ I S Gi for all G then there exists a finite subset J ⊆ I such that [[ ~x . φ ]] G = S i ∈ J S Gi . Thefact that the classifying topos of G p is coherent also implies, by Deligne’stheorem (assuming the axiom of choice), the existence of set-based mod-els for any non-contradictory quotient of G p whose associated Grothendiecktopology on f.g. Grp op is of finite type. Let A be the algebraic theory of groups. Since every finite group is finitelypresented as a A -model (cf. Theorem 6.4 [6]) and A is of presheaf type,Theorem 6.29 ensures that there exists a quotient U of A classified by thetopos [ C , Set ] , where C is the category of finite groups and homomorphismsbetween them, which can be characterized as the set of all geometric sequentsover the signature of A that are valid in every finite group. On the otherhand, since U is classified by the topos [ C , Set ] , the set-based models of U are exactly the groups which can be expressed as filtered colimits of finitegroups, such groups are exactly the groups which validate all the geometricsequents over the signature of A which hold in every finite group. As aby-product, we obtain the following characterization of locally finite groups(equivalently, of the groups which can be expressed as filtered colimits offinite groups). Proposition 8.2.
The locally finite groups are exactly the groups which sat-isfy all the geometric sequents over the signature of the theory of groups whichhold for all finite groups.
Proof
In view of the above remarks, it remains to verify that a group islocally finite (in the sense that all its finitely generated subgroups are finite)if and only if it is a filtered colimit of finite groups. This can be provedas follows. If a group is locally finite then it is the filtered union of all itsfinitely generated (and hence finite) subgroups. Conversely, suppose that G is a filtered colimit of finite groups. Then G is the directed union of the154mages in G of these finite subgroups, which are again finite. It follows thatevery finitely generated subgroup H of G is contained in one of them (noticethat, since the union is filtered, there exists one of them which contains allthe generators of H ) and hence it is a fortiori finite, as required. (cid:3) The injectivization of A is also of presheaf type (by Corollary 6.57) andcan be characterized as the quotient of G consisting of all the geometricsequents over the signature of the injectivization G of the theory of groupswhich hold in all finite groups. Let V K be the expansion of the algebraic theory of vector spaces over afield K obtained by adding a n -ary predicate R n for each natural number n expressing the property of a n -tuple of elements to be linearly independent,i.e. the following sequents: ( ⊤ ⊢ ~x R n ( ~x ) ∨ _ ( k ,...,k n ) ∈ K n k i =0 for some i k x + · · · + k n x n = 0) , and ( R n ( ~x ) ∧ ( _ ( k ,...,k n ) ∈ K n k i =0 for some i k x + · · · + k n x n = 0) ⊢ ~x ⊥ ) . The category of models of the theory V K in Set has as objects the vectorspaces over K and as arrows the injective homomorphisms between them.The finitely presentable V K -models are precisely the finite-dimensional vec-tor spaces over K .By using techniques analogous to those employed in section 8.2, one canprove that V K is of presheaf type.Also, by using Corollary 6.22, one can easily prove that for any fixednatural number n , both the expansion of the theory of vector spaces over afield K and of the theory V K obtained by adding the sequent ⊤ ⊢ ( x ,...,x n +1 ) _ ( k ,...,k n +1 ) ∈ K n +1 k i =0 for some i k x + · · · + k n +1 x n +1 = 0) are of presheaf type.The models in Set of the latter theory are precisely the vector spacesover K of dimension ≤ n . l -groups with strong unit Recall that an abelian l -group with strong unit is a lattice-ordered group ( G, , ≤ ) with a distinguished element u , called the unit of the group, such155hat for any element x ∈ G such that x ≥ there exists a natural number n such that x ≤ nu , where nu = u + · · · + u n times. We refer the reader tochapter of [31] as an introduction to the theory of lattice-ordered groups.We can axiomatize the theory L u of abelian l -groups with strong unit overa signature Σ consisting of four binary function symbols + , − , inf , sup , twoconstants and u and a binary relation symbol ≤ , by using Horn sequents toformalize the notion of abelian l -group and the following geometric sequentto express the property of strong unit: ( x ≥ ⊢ x _ n ∈ N x ≤ nu ) . The following lemma will be useful in showing that the theory L u is ofpresheaf type. Lemma 8.3.
Let G be an abelian group with a distinguished element u andgenerators x , . . . , x n . If for every i ∈ { , . . . , n } there exists a natural num-ber k i such that | x i | ≤ k i u then u is a strong unit for G . Proof
Recall that the absolute value | x | of an element x of an abelian l -group with unit ( G, , + , − , ≤ , inf, sup ) is the element sup ( x, − x ) . For any x ∈ G , | x | ≥ | x | = | − x | , and for any x, y ∈ G , the triangular inequality | x + y | ≤ | x | + | y | holds.Since G is generated by elements x , . . . , x n , every element x of G can beexpressed as the interpretation t ( x , . . . , x n ) of a term t over the signature Σ . We shall prove that there exists a natural number n such that | x | ≤ nu by induction on the structure of t . This will clearly imply our thesis, sinceif x ≥ then | x | = x . If t is a variable then the claim is clearly true byour hypothesis. If x = x ′ + x ′′ with | x ′ | ≤ n ′ u and | x ′′ | ≤ n ′′ u then by thetriangular inequality we have | x | ≤ n ′ + n ′′ , and similarly for the subtraction.The inf and sup cases are similarly straightforward. (cid:3) Let us now verify that the theory L u satisfies the hypotheses of Corollary6.22 with respect to its Horn part H .We have to prove that for any finitely presentable H -model c , any model G of L u in a Grothendieck topos E , any object E of E and any Σ -structurehomomorphism f : c → Hom E ( E, G ) , there exists an epimorphic family { e i : E i → E | i ∈ I } in E and for each i ∈ I a finitely presentable model c i of L u and Σ -structure homomorphisms f i : c → c i and u i : c i → Hom E ( E i , G ) suchthat Hom E ( e i , G ) ◦ f = u i ◦ f i for all i ∈ I . Let us suppose that c is presentedas a H -model by a cartesian formula φ ( ~y ) with generators x , . . . , x n . Since G is a l -group with strong unit, there exists an epimorphic family { e i : E i → E | i ∈ I } in E and for each k ∈ { , . . . , n } and i ∈ I a natural number m k,i such that f ( | x i | ) ◦ e i ≤ m k,i u Hom E ( E i ,G ) (where u Hom E ( E i ,G ) denotes the unitof the ℓ -group Hom E ( E i , G ) ). For each i ∈ I , let c i the H -model presented156y the cartesian formula φ ( x , . . . , x n ) ∧ | x | ≤ m ,i ∧ . . . ∧ | x n | ≤ m n,i . Foreach i ∈ I , we have a natural quotient homomorphism f i : c → c i throughwhich Hom E ( e i , G ) ◦ f factors; the resulting factorization u i satisfies therequired property Hom E ( e i , G ) ◦ f = u i ◦ f i . Since H is a Horn theory, each c i is generated by the n -tuple f i ( x ) , . . . , f i ( x n ) which presents it as a H -model (cf. Remark 4.2(a)). Therefore the c i are l -groups with strong unitby Lemma 8.3.This argument also shows that the finitely presentable L u -models areexactly the finitely presented H -models whose unit is strong (cf. Theorem6.26).Therefore all the hypotheses of Corollary 6.22 are satisfied and we canconclude that the theory L u is of presheaf type. In fact, L u is Morita-equivalent to the (algebraic) theory of MV-algebras (cf. [12]). References [1] M. Artin, A. Grothendieck and J. L. Verdier,
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