aa r X i v : . [ m a t h . C T ] J a n On Supercompactly and Compactly Generated Toposes
Morgan Rogers ∗†‡ th January, 2021
Abstract
We present and characterize the classes of Grothendieck toposes having enoughsupercompact objects or enough compact objects. In the process, we examine thesubcategories of supercompact objects and compact objects within such toposesand classes of geometric morphism which interact well with these objects. We alsopresent canonical classes of sites generating such toposes.
Contents ∗ Universit`a degli Studi dell’Insubria, Via Valleggio n. 11, 22100 Como CO † Marie Sklodowska-Curie fellow of the Istituto Nazionale di Alta Matematica ‡ email: [email protected] ibliography 60 Introduction
In [20], the author of the current paper obtained a characterisation of toposes ofright actions of discrete monoids on sets, which are a special case of presheaf toposes.A natural next step in this direction is to consider sets as discrete spaces and toexamine the categories of actions of topological monoids on them. It turns out thatthe resulting categories, to which a future paper will be devoted, are also toposes,and that moreover they fall into the class of supercompactly generated toposes whichwe present and thoroughly investigate in this paper.By a supercompactly generated topos, we mean a topos with a separating setof supercompact objects (see Definition 1.1). Apart from the toposes of topologicalmonoid actions mentioned above, this class includes all regular toposes (Example1.5), as well as all presheaf toposes and some other important classes described inProposition 1.6, and so is of general interest in topos theory. Since it is convenientto do so, we also study compactly generated toposes , which are conceptually similarenough that we can prove analogous results about them with little extra work.For brevity, some of the section headings refer only to the supercompact namingconventions.This is not the first time these concepts have been studied. Some relevant resultsappear in Section 4.1 of the thesis of Bridge [2], where our supercompactly generatedtoposes are referred to as ‘ locally supercompact ’; the reason for the terminologychoice of the present author is that the adverb “locally” is already overloaded intopos-theory literature. Moreover, a special case of principal sites, ‘ B -sites ’, are theeventual focus of the paper [17] of Kondo and Yasuda. We posit that where thereare overlaps in the basic results of the present paper and those above (which weshall point out in situ), the main distinction of the present paper is the emphasis ontopos-theoretic machinery: the present author avoids reasoning directly with sheavesas far as possible (with the exception of the non-constructive results of Section 2.7),which allows for more concise categorical proofs.We assume the reader is familiar with the basics of category theory and Grothendiecktoposes; the content of Mac Lane and Moerdijk’s textbook [18] should be adequate.For Section 2 we rely heavily on the recent monograph of Caramello [8] which con-tains general results about site representations of toposes and morphisms of sites;we quote some of those results without proof here.This work was supported by INdAM and the Marie Sklodowska-Curie Actionsas a part of the INdAM Doctoral Programme in Mathematics and/or ApplicationsCofunded by Marie Sklodowska-Curie Actions . The author would like to thank OliviaCaramello for her patience and helpful suggestions.
Overview
The structure of this paper is as follows. In Section 1.1, we recall the definitionsof supercompact and compact objects in a topos. This leads us to formally definesupercompactly and compactly generated toposes in Section 1.2.In Section 1.3, we turn to the full subcategories of supercompact and compactobjects in a topos, presenting the structure they inherit from their ambient toposes,with a focus on monomorphisms, epimorphisms, and the classes of funneling and ultifunneling colimits, which we introduce in Definitions 1.9 and 1.11. The purposeof this investigation is to present these subcategories as canonical sites for super-compactly and compactly generated toposes in Section 1.4, which we do in Theorem1.23. Motivated by the special case of toposes of monoid actions, in Section 1.5we investigate some extra conditions on a supercompactly generated topos whichguarantee further properties of its category of supercompact objects.In Section 1.6, we examine classes of geometric morphism whose inverse imagefunctors preserve supercompact or compact objects, introducing notions of precise and polite geometric morphisms in analogy with proper geometric morphisms (Def-inition 1.34), before focussing on relative versions of these properties (Definition1.36) which are more directly useful in our analysis. After establishing these defi-nitions, we examine how more familiar classes of geometric morphism interact withsupercompactly and compactly generated toposes: surjections and inclusions in Sec-tion 1.7, then hyperconnected morphisms in Section 1.8. This exploration gives usseveral tools for constructing such toposes, which are summarised in Theorem 1.54.The focus of Section 2 is a broader site-theoretic investigation. In Section 2.1,we exhibit the categorical data of principal and finitely generated sites, which arenatural classes of sites whose categories of sheaves are supercompactly and com-pactly generated toposes, respectively. In Section 2.2, we examine the morphismsbetween the representable sheaves on these sites, and then show in Section 2.3 howa general such site may be reduced via a canonical congruence without changingthe resulting topos. We use what we have learned about these sites in Section 2.4to characterize the categories of supercompact and compact objects which were thesubject of Section 1.4 as reductive and coalescent categories, respectively (Definition2.23), satisfying additional technical conditions. We make the correspondence be-tween such categories and the toposes they generate explicit in Theorem 2.32. It isnatural to compare these classes of categories to the well-known classes of (locally)regular and coherent categories, which we do in Section 2.5.Moving onto morphisms, we recall the definition of morphisms of sites in Section2.6, showing that, according to the class of sites under consideration, these inducethe relatively precise, polite or proper geometric morphisms introduced in Section1.6. More significantly, restricting to canonical sites, we are able to extend thecorrespondences of Theorem 2.32 to some 2-equivalences between 2-categories of sitesand 2-categories of toposes. In Section 2.7, we examine the points of supercompactlyand compactly generated toposes, extending the classical result of Deligne to showthat any compactly generated topos has enough points. To ground the discussion,we present some examples and counterexamples of reductive and coalescent sitesand their properties in Section 2.8.Finally, in Section 3 we examine the special case of supercompactly and com-pactly generated localic toposes, recovering some Stone-type dualities and coun-terexamples in the process. Throughout, when ( C , J ) is a site, we write ℓ : C →
Sh( C , J ) for the composite ofthe Yoneda embedding and the sheafification functor, assuming the Grothendiecktopology is clear in context, and call the images ℓ ( C ) of the objects C ∈ C the representable sheaves . .1 Supercompact Objects The following definitions can be found in [7, Definition 2.1.14]:
Definition 1.1.
An object C of a category E is supercompact (resp. compact )if any jointly epic family of morphisms { A i → C | i ∈ I } contains an epimorphism(resp. a finite jointly epic sub-family).Clearly every supercompact object is compact. Compact objects are more widelystudied, notably in the lifting of the concept of compactness from topological spacesto toposes reviewed by Moerdijk and Vermeulen in [19]. Since the two classes ofobjects behave very similarly, we treat them in parallel.In a topos, we may re-express the definitions of supercompact and compactobjects in terms of their subobjects. As is standard, we can further convert anystatement about subobjects of an object X in a topos E into a statement about thesubterminal object in the slice topos E /X . Lemma 1.2.
An object C of a Grothendieck topos E is supercompact (resp. compact)if and only if every covering of C by a family (resp. a directed family) of subobjects A i ֒ → C contains an isomorphism. This occurs if and only if the global sectionsfunctor Γ : E /C → Set preserves arbitrary (resp. directed) unions of subobjects.Proof.
For the first part, suppose that we are given a family (resp. a directedfamily) of subobjects covering a supercompact (resp. compact) object C . Then oneof the monomorphisms involved must be epic and hence an isomorphism (resp. thiscollection contains a finite covering family, but the union of these subobjects is alsoa member of the family and must be covering). In the opposite direction it sufficesto consider images of the morphisms in an arbitrary covering family (resp. finiteunions of these images).The remainder of the proof is modelled after that of Johnstone in [16, C1.5.5].Given an object f : A → C of E /C which is a union (resp. a directed union) ofsubobjects A i ֒ → A → C and given a global section x : C → A of f , we may considerthe pullbacks: C i CA i A. y x By extensivity, C is the union of the C i , and by the above one of the C i ֒ → C mustbe an isomorphism, so that x factors through one of the A i , which gives the result.Conversely, given a jointly epic family (resp. directed family) of subobjects C i ֒ → C considered as subterminals in E /C , we may apply Γ to see that one of themmust be an isomorphism, as required. The collections of supercompact and compact objects in any
Grothendieck toposare conveniently tractable:
Lemma 1.3.
Let
E ≃
Sh( C , J ) be a Grothendieck topos of sheaves on a small site ( C , J ) . Then the supercompact objects are quotients of the representable sheaves ℓ ( C ) for C ∈ C . In particular, they are indexed (up to isomorphism) by a set. Similarly, he compact objects are quotients of finite coproducts of the images ℓ ( C ) , and so (upto isomorphism) also form a set.Proof. Given a supercompact object Q , since the objects ℓ ( C ) are separating in E ,the collection of morphisms ℓ ( C ) → Q (is inhabited and) jointly epimorphic. Itfollows that one such must be epimorphic.Given a compact object Q , the above argument instead yields a (possibly empty)finite jointly epimorphic family of morphisms ℓ ( C i ) → Q , which corresponds to anepimorphism ` i ∈ I ℓ ( C i ) ։ Q , as claimed.Lemma 1.3 ensures that we can always consider the full subcategories on the su-percompact (resp. compact) objects, equipped with the canonical topology inducedby the topos (which shall be recalled in Definition 1.21 below) as an essentiallysmall site ( C s , J E can | C s ) (resp. ( C c , J E can | C c )). By the Comparison Lemma, the in-duced canonical comparison morphism E →
Sh( C s , J E can | C s ) is an equivalence if andonly if the collection of supercompact objects is separating, and similarly for thecompact case. We shall continue to use C s and C c to denote these categories inthe remainder; for simplicity, we shall actually assume that we have chosen a rep-resentative set of the supercompact or compact objects, such as the quotients ofrepresentables in Lemma 1.3, so that we are working with small sites. Definition 1.4.
We say a topos is supercompactly generated (resp. compactlygenerated ) if its collection of supercompact (resp. compact) objects is separating.
Example 1.5.
The syntactic category of a regular theory can be recovered from itsclassifying topos as the full category of regular objects. In general, we say that anobject X in a topos is regular if X is supercompact and for any cospan Y X Z f g with Y and Z supercompact, the pullback Y × X Z is also supercompact. In par-ticular, classifying toposes of regular theories are special cases of supercompactlygenerated toposes. The same can be said when ‘supercompact’ is replaced by ‘com-pact’ and ‘regular’ is replaced by ‘coherent’. See the work of Caramello [5] for thisresult and a more detailed discussion (note that Caramello refers to regular objectsas supercoherent objects). We shall return to examination of categories of regularand coherent objects in Section 2.5.Supercompactly generated toposes include several other important establishedclasses of Grothendieck topos. Proposition 1.6. (i) Every atomic topos is supercompactly generated.(ii) Every supercompactly generated topos is compactly generated and locally con-nected.(iii) Every presheaf topos is supercompactly generated.Proof.
For (i), recall that a Grothendieck topos is atomic if and only if it has aseparating set of atoms, and these are easily seen to be supercompact. For (ii), wecan similarly observe that any supercompact object is compact and indecomposable,then recall (by Theorem 2.7 of [5], say) that a topos is locally connected if and onlyif it has a separating set of indecomposable objects. or (iii), note that the representable presheaves are irreducible (every jointly epicfamily over a representable contains a split epimorphism) so they are in particularsupercompact. We now examine properties of the categories C s and C c in a general topos E . Lemma 1.7.
Let E be a Grothendieck topos and let C s , C c be the categories ofsupercompact and compact objects of E respectively. These categories are closed in E under quotients.Proof. Given an epimorphism k : D ։ C with D supercompact and a coveringfamily over C , pulling back this family along k we immediately conclude that oneof the constituent morphisms must be an epimorphism. Thus C is a member of C s .The argument for C c is analogous, except that we end up with a finite family ofmorphisms.Lemma 1.7 has as a consequence that when considering a covering family ofsupercompact or compact objects over an object X of E , we may without loss ofgenerality assume that the morphisms in the family are monomorphisms. That is,we may restrict attention to covering families of (super)compact subobjects whenwe so choose, because a family of morphisms with common codomain in a topos isjointly epic if and only if the union of their images is the maximal subobject.The subcategories inherit some further structure from E . Corollary 1.8.
For E , C s , C c as in Lemma 1.7, C s and C c are closed under imagefactorizations in E , so that in particular they have image factorizations.Proof. Given a morphism C → C ′ between supercompact (resp. compact) objects,the image object C ′′ in the factorization C ։ C ′′ ֒ → C ′ is also supercompact (resp.compact) by Lemma 1.7, whence the factoring morphisms lie in C s (resp. C c ) sinceit is a full subcategory.Note that the resulting orthogonal factorization systems on C s and C c are notbetween all monomorphisms and all epimorphisms; only between those inheritedfrom E . We spend the rest of this section deriving an intrinsic characterisation ofthese morphisms. Definition 1.9.
We say a small indexing category D is a funnel if it has a weaklyterminal object. A funneling diagram in an arbitrary category C is a functor F : D → C with D a funnel. For example: A i ... D.A j f i f ′ i f j f ′ j The colimit of F , if it exists, is an object C of C equipped with an epimorphism f : D ։ C through which all legs of the colimit cone factor. ecall that a morphism h : D → C in a category C is called a strict epimor-phism if whenever another morphism k : D → E satisfies the condition that foreach parallel pair p, q : B ⇒ D with h ◦ p = h ◦ q we have k ◦ p = k ◦ q , it followsthat k factors uniquely through h . The dual concept appears in [6, Theorem 4.1]. Lemma 1.10.
In a small category C , a morphism h : C ′ → C is a strict epimor-phism if and only if there exists a funneling diagram F : D → C with weakly terminalobject C ′ whose colimit is expressed by h .Proof. By definition, if h is a strict epimorphism, it is a colimit for the diagramconsisting of all pairs of morphisms with domain C ′ which h coequalizes. Conversely,if h expresses the colimit of any funneling diagram, and k coequalizes all of the sameparallel pairs that h does, then it clearly induces a cone by composition with themorphisms of the funneling diagram, whence it has a universal factorisation through h , as required.Notably, strict epimorphisms include isomorphisms and regular epimorphisms.Continuing with the parallel treatment of compactly generated toposes, we arriveat the following definitions. Definition 1.11.
A small indexing category D is a multifunnel if it has a (possi-bly empty) finite collection of objects D , . . . , D n to which all other objects admitmorphisms . A colimit of a multifunneling diagram (a diagram indexed by a mul-tifunnel) in a category C shall be called a multifunneling colimit , and is definedby a finite jointly epic family from the images of the objects D , . . . , D n . A finitejointly epic family obtained in this way will be called a strictly epic finite family . Lemma 1.12.
A category has multifunneling colimits if and only if it has finitecoproducts and funneling colimits.Proof.
Clearly finite coproducts and funneling colimits are special cases of multi-funneling colimits. Conversely, given a multifunneling diagram F : D → C withweakly terminal objects F ( D ) , . . . , F ( D n ), consider the coproduct of these objects.Composing the morphisms in the diagram F with the coproduct inclusions, we get afunneling diagram. The universal property of the coproduct ensures that the colimitof this diagram coincides with the colimit of F . Remark . Note that we can make the further simplification, implicit in thediagram of Definition 1.9, that all of the non-identity morphisms in a funnelingdiagram have the weakly terminal object as their codomains, since given t : A i → A j ,there exists some morphism f j : A j → D , and the cocone commutativity conditionsfor f j ◦ t and f j ensure that λ i = λ D ◦ ( f j ◦ t ) = λ j ◦ t is automatically satisfied, sowe may omit t from the diagram. Lemma 1.14.
Let E be a Grothendieck topos and let C s , C c be the categories ofsupercompact and compact objects of E respectively. Then C s is closed in E underfunneling colimits and C c is closed in E under multifunneling colimits. A reader interested in the obvious generalisations of this concept to higher cardinalities might preferto employ a name such as ‘finitely funneled’ to emphasise the finitary aspect. roof. Let F : D → C s be a funneling diagram with weakly terminal object F ( D ).In E , this diagram has a colimit determined by an epimorphism f : F ( D ) ։ C ; thecolimit C is supercompact by Lemma 1.7, as required. The argument for multifun-neling colimits in C c is analogous, except that we must pull back along each memberof a finite family in the proof of Lemma 1.7 to obtain the finite covering subfamilyof a given covering family over the colimit. Corollary 1.15.
Let E be a supercompactly (resp. compactly) generated topos and C s , C c the usual subcategories. Then a morphism of C s is an epimorphism in E ifand only if it is a strict epimorphism in C s , and a finite family of morphisms into C in C c is jointly epic in E if and only if it is a strictly epic finite family in C c .Proof. This could be deduced by checking the conditions of the dual of [6, Proposi-tion 4.9], but rather than reproducing that result, we give a direct proof.Suppose h : C ′ ։ C is epic in E with C ′ , C in C s . Any epimorphism in E isregular, so is the coequalizer of some pair p, q : D ⇒ C ′ . Since D is covered bysupercompact objects, composing p and q with the monomorphisms C i ֒ → D suchthat C i is in C s we obtain a funneling diagram in C s whose colimit is still C , asrequired. The argument for C c is analogous.Conversely, the inclusion of C s and C c into E preserves funneling (resp. multi-funneling) colimits by Lemma 1.14, whence the strict epimorphisms (resp. strictlyepic finite families) from these categories are still epic in E .In fact, epimorphic families in C c are better behaved than those in C s in general: Lemma 1.16.
Let E be any Grothendieck topos. Then a family of morphisms in C c with common codomain is jointly epic in C c if and only if it is so in E . In particular,when E is compactly generated, every jointly epimorphic family (including everyepimorphism) in C c is strict.Proof. Observe that a family of morphisms f i : C i → C in a category is jointlyepimorphic if and only if the diagram: C i C j CC C . . . f i f i f j f j is a colimit diagram. But the diagram (after removing the copy of C in the lowerright corner) is clearly an instance of a multifunneling colimit, so by Lemma 1.14its colimit is created by the inclusion of C c into E , so a family is jointly epic in C c ifand only if it is so in E , as required.Having extensively discussed the epimorphisms, we should also discuss monomor-phisms in the subcategories under investigation. Lemma 1.17.
Monomorphisms in C s and C c coincide with monomorphisms in E when E is supercompactly or compactly generated, respectively. roof. Certainly a monomorphism of E lying in C s or C c is still monic, since thereare fewer morphisms which it needs to distinguish in general.Suppose E is supercompactly generated and let s : A ֒ → B be a monomorphismin C s , Q an object of E and f, g : Q ⇒ A such that sf = sg . Covering Q withsupercompact subobjects q i : Q i ֒ → Q , consider f q i , gq i : Q i ⇒ A , which aremorphisms in C s . These are equalized by s and hence are equal for every i . The q i being jointly epic then forces f = g . Thus s is monic in E , as claimed. The argumentfor C c is analogous, replacing supercompact subobjects with compact ones.Once again, we can immediately strengthen this result for C c . Lemma 1.18.
For any Grothendieck topos E , the monomorphisms of E lying in C c are regular monomorphisms there. In particular, when E is compactly generated,every monomorphism in C c is regular.Proof. If e : C ′ ֒ → C is a morphism in C s which is monic in E , consider its cokernelpair in E : C ′ CC D. ee y st p Since a topos is an adhesive category, this is also a pullback square and so e is theequalizer of s and t . Since pushouts are multifunnel colimits, D lies in C s , so thesame is true there.Thus we can make Corollary 1.8 more precise. Corollary 1.19. If E is supercompactly generated, then C s has an orthogonal (strictepi,mono)-factorisation system. More generally, if E is merely compactly generated, C c has an orthogonal (epi,mono)-factorisation system. For later reference, we observe that even though the categories C s and C c neednot have finite products (see Example 2.60), we can still extend Corollaries 1.8 and1.19 with factorizations of spans through jointly monic spans. Corollary 1.19 is thecase I = 1 of the following Lemma. Lemma 1.20.
Let { f j : B → A j | j ∈ I } be a collection of morphisms withcommon domain in C s or C c . Then there exists a strict epimorphism e : B ։ R andmorphisms { r j : R → A j | j ∈ I } which are jointly monic, such that f j = r j ◦ e .Proof. Let e be the strict epimorphism obtained from the funneling colimit of thecollection of all parallel pairs of morphism which are coequalized by all of the f j .By definition, all of the f j factorize through it, and by construction the factors forma jointly monic family. We have now done enough work to usefully apply the proof of Giraud’s theorem andobtain a canonical site of definition for a supercompactly or compactly generatedGrothendieck topos. efinition 1.21. Recall that a sieve S on an object C of a category C is effective-epimorphic if, when S is viewed as a full subcategory of C /C , C is the colimit ofthe (possibly large) diagram D S : S ֒ → C /C → C obtained by composing with theforgetful functor. A sieve generated by a single morphism f is effective-epimorphicif and only if the morphism is a strict epimorphism. Such a sieve S is universally effective-epimorphic if its pullback along the functor C /D → C /C induced by amorphism f : D → C is effective-epimorphic for any f .The canonical Grothendieck topology J C can on C is the topology whosecovering sieves are precisely the universally effective-epimorphic ones. If C is aGrothendieck topos, this coincides with the Grothendieck topology whose coveringsieves are those containing small jointly epic families.We recall the following result, which appears as [8, Proposition 4.36]: Lemma 1.22.
Let E be a Grothendieck topos and C a small full separating subcat-egory of E . Let S be a sieve in C on an object C and let D S be the diagram in C described in Definition 1.21. Suppose that the colimit of D S in E lies in C . Then S is universally effective-epimorphic in C if and only if it is the restriction to C of asieve containing a small jointly epic family in E . Theorem 1.23.
Suppose E is supercompactly generated. Let J r be the Grothendiecktopology on C s whose covering sieves are those containing strict epimorphisms. Then E ≃
Sh( C s , J r ) .Similarly, if E is compactly generated, and J c is the Grothendieck topology on C c whose covering sieves are those containing strictly epic (equivalently, jointly epic)finite families. Then E ≃
Sh( C c , J c ) .Proof. By Giraud’s theorem, given a (small, full) separating subcategory C of objectsin a Grothendieck topos E , we have an equivalence of toposes E ≃
Sh( C , J E can | C ),where J E can | C is the restriction of the canonical topology on E to C , whose coveringsieves are the intersections of J E can -sieves with C . Thus it suffices to show in eachcase that the restriction of the canonical topology is the topology described in thestatement.By Lemma 1.14, the principal (resp. finitely generated) sieves S on C s (resp. C c ),whose corresponding diagrams D S are funneling (resp. multifunneling) diagrams,have colimits contained in C s (resp. C c ), so Lemma 1.22 applies. Thus these areeffective epimorphic sieves if and only if the generating morphism is a strict epi-morphism in C s (resp. the generating morphisms form a strictly epimorphic finitefamily in C c ).Now given any sieve S containing a jointly epic family on an object of C s (resp. C c ) in E , by the definition of supercompact (resp. compact) objects, S must containan epimorphism (resp. a finite covering family). In particular, every J E can | C s -coveringsieve contains a J E can | C s -covering principal sieve. Similarly, every J E can | C c -coveringsieve contains a J E can | C c -covering finitely generated sieve.It follows that the strict epimorphisms in C s are precisely the morphisms gen-erating universally effective-epimorphic families (and similarly for strict jointly epi-morphic families in C c ), as required. Alternatively, see the proof of Proposition 2.8below for a direct argument showing that the strict epimorphisms (resp. strictlyepic finite families) are stable. emark . It should be clear by now from our joint treatment of supercompact-ness and compactness that much of our analysis can be applied to more general no-tions of compactness. Indeed, Theorem 1.23 is an explicit special case of Caramello’s[8, Proposition 4.36].Suppose P is some property of pre-sieves (families of morphisms with commoncodomain); then we may define P -compact objects in a topos E as those for whichevery jointly epic covering family contains a jointly epic presieve satisfying P . If P satisfies suitable composition and stability criteria, which Caramello specifies, andthe full subcategory C P of E on the P -compact objects is separating and closed undercertain colimits, then E is equivalent to the category of sheaves on C P for the topologygenerated by the effective-epimorphic P -presieves in C P . For supercompactness, P is the property ‘is a singleton’, while for ordinary compactness, P is the property ‘isfinite’, and our earlier results show that these do satisfy Caramello’s criteria.While we shall not attempt to extend the present paper to this most generalcase, we encourage the reader to explore whether any given topos of interest tothem is P -compactly generated for some suitable property P , and if so to computethe corresponding site produced by Caramello’s result.The advantage of the intrinsic expressions for the Grothendieck topologies inTheorem 1.23 is that it guarantees that the categories of (super)compact objectscontain enough information to completely reconstruct the toposes by themselves .This immediately gives us results such as the following: Corollary 1.25.
Suppose E and E ′ are supercompactly generated toposes and C s , C ′ s are their respective categories of supercompact objects. Then E ≃ E ′ if and onlyif C s ≃ C ′ s . We saw in Lemmas 1.16 and 1.18 that when E is compactly generated, every epi-morphism in C c is strict and every monomorphism in C c is regular. This leads usto wonder under what extra conditions these facts hold true in C s , given that E issupercompactly generated. Example 1.26.
To properly motivate this section, we show that epimorphisms in C s need not coincide with those in E . Let D be the category A B C. l r
In the topos E of presheaves on D it is easily calculated that the supercompactobjects are precisely the representables, so D coincides with C s (we shall see inProposition 3.1 that this argument is valid for all posets). The morphisms l and r are trivially epic in D but are not epic in E .Our main tool in this section is the following definition. Definition 1.27.
Given a morphism f : A → B , its cokernel B → B/f is thepushout:
A B B/f. f ! x p Not to be confused with the cokernel pairs mentioned in the proof of Lemma 1.18. okernels are useful for understanding epimorphisms thanks to the followingresult. Lemma 1.28.
A morphism f : A → B of a topos E is an epimorphism if and only ifthe lower morphism x : 1 → B/f of its cokernel is an isomorphism (or equivalently,an epimorphism).Proof. If f is an epimorphism we have: A B B/f, f ! x p since the pushout of an epimorphism is epic. But any quotient of 1 in a topos is anisomorphism, as required.Conversely, if x : 1 → B/f is an isomorphism, we can consider the epi-monofactorization f = m ◦ e : A A ′ B B/f. fe ! m ! x ∼ ∼ p By the first part, the left hand square and outside rectangle are both pushouts,which makes the right hand square a pushout. But in a topos (or any adhesivecategory), a pushout square in which the upper horizontal morphism is monic isalso a pullback square. Thus m is an isomorphism, and f is epic.Recall that an object A of a category with a terminal object is well-supported if the unique morphism ! A : A → E . Using cokernels, weobtain a partial dual to Lemma 1.17 even without requiring E to be (super)compactlygenerated. Lemma 1.29.
Let E be a topos and C s the usual subcategory. Let A be an object of C s which is well-supported as an object of E . Then a morphism A → B of C s is anepimorphism in that category if and only if it is an epimorphism in E .Proof. An epimorphism of E lying in C s clearly remains epic there.Conversely, if e : A ։ B is epic in C s , consider the cokernel B → B/e : A B B/e. e ! A qx p By assumption
B/e is supercompact as q is epic in E . The unique morphism ! A : A → e via the unique morphism ! B : B →
1. Thus we have qe = x ! A = x ! B e , and since these expressions are composed of morphisms lying in C s where e is epic, it follows that q = x ! B whence x is epic and hence an isomorphism.Hence e is epic in E , by Lemma 1.28. t is worth noting that the existence of any well-supported supercompact objectforces the terminal object of E to be supercompact. Moreover, this proof concertedlyfails when the object A is not well-supported: Lemma 1.30.
Let E be a supercompact topos. Then C s is closed under cokernels ifand only if every supercompact object is well-supported.Proof. Given a supercompact object A , consider its support , which is the subter-minal object U ֒ → A . By Lemma 1.7, U is supercompact, so if C s is closed under cokernels, the pushout of U ֒ → C s . But the colimit morphisms 1 ⇒ U U ∼ = 1.Thus A is well-supported, as required.Conversely, if every object A of C s is well-supported then the cokernel (in E ) ofa morphism A → B in C s is a quotient of B and so is supercompact. Thus C s isclosed under cokernels, as required.We shall see a relevant sufficient condition for this to occur in Proposition 1.49.In this setting we can also strengthen Lemma 1.17. Scholium 1.31.
Let E be a topos such that every object of C s is well-supported.Then monomorphisms in C s inherited from E are regular. In particular, if E is alsosupercompactly generated, then all monomorphisms in C s are regular.Proof. By Proposition 1.49, the hypotheses guarantee that the cokernel of a mor-phism in C s also lies in that category.As remarked in the proof of Lemma 1.28, the pushout square defining the cokernelof an inclusion of supercompact objects i : A ֒ → B is also a pullback in E . It followseasily that i is the equalizer in E of the morphisms q, x ! B : B ⇒ B/i described inthe proof of Lemma 1.29, and is consequently also their equalizer in C s .If E is supercompactly (resp. compactly) generated then we may apply Lemma1.17 to conclude that the above applies to all monomorphisms of C s (resp. C c ).On the other hand, we shall see in Example 1.51 that it is not in general possibleto strengthen the properties of epimorphisms in C s beyond the consequence of theproof of Theorem 1.23 that they are strict. In particular, they are not regular ingeneral.Finally, we explicitly record how Lemma 1.29 sjimplifies the expression for theGrothendieck topology J r induced on C s from Theorem 1.23. Corollary 1.32.
Let E be a supercompactly generated topos and C s its full subcat-egory of supercompact objects. Suppose every object of C s is well-supported. Let J r be the topology on C s whose covering sieves are precisely those containing epimor-phisms. Then E ≃
Sh( C s , J r ) . In this section we present some classes of geometric morphism whose inverse imagefunctors interact well with supercompact and compact objects. The material in thissection contains more technical topos theoretic concepts than that in the earliersections, so may be skipped on a first reading; the key result is Proposition 1.40. he latter part of Lemma 1.2, regarding compact objects, states precisely thatthe geometric morphism E /C → Set is proper in the sense of Moerdijk and Ver-meulen [19, Definition I.1.8] (see also Johnstone [16, C3.2.12 and C1.5.5]), or equiv-alently that E /C is a compact Grothendieck topos. We can generalise Moerdijk andVermeulen’s definition of proper morphisms to the arbitrary union case in order tocapture the idea of supercompactness. In order to do this, we recall some classictopos-theoretic constructions, from [14, Chapter 2].Recall from [14, Definition 2.11] that an internal category I in a topos (or,more generally, a cartesian category) E consists of:(i) An objects of objects I and an object of morphisms I in E , and(ii) Morphisms i : I → I , d, c : I ⇒ I and m : I → I ,where I is the object of composable pairs , defined as the pullback: I I I I . π π y dc The morphisms define the identity morphisms , domains , codomains and composition ,respectively. These are required to satisfy the equations di = ci = id I , dm = dπ , cm = cπ , m (id × m ) = m ( m × id) and m (id × i ) = m ( i × id) = id I . These arediagrammatic translations of the axioms for ordinary categories in Set . An internalfunctor between internal categories is a pair of morphisms between the respectiveobjects of objects and objects of morphisms commuting with the respective structuremorphisms.
Definition 1.33.
We say an internal category in E is inhabited if its object ofobjects is well-supported. An internal category is directed if it is moreover directedin the internal logic of E (which implies inhabitedness).Given an internal category I in E , we recall from [14, Definition 2.14] that an(internal) diagram of shape I consists of:1. An object a : F → I of E /I , and2. A morphism b : F → F ,where F is the defined as the pullback F F I I , π π y ad such that ab = cπ , b (id × i ) = id F , and e ( e × id) = e (id × m ). In Set , this datacaptures the encoding of a functor into
Set via the Grothendieck construction.There is an accompanying notion of natural transformation, and hence we obtainthe internal diagram category [ I , E ]. This is a topos over E , which is proved usingthe comonadicity theorem in [14, Corollary 2.33]. This is the constructive term for ‘non-emptiness’, meaning ‘has an element’. Translating this condi-tion into the internal language of the topos E , we get the well-supportedness condition. efinition 1.34. For every object E of E and every internal category I (resp.inhabited internal category I ; filtered internal category I ) in E /E , there is an inducedpullback square of diagram toposes :[ f ∗ ( I ) , F /f ∗ ( E )] [ I , E /E ] F /f ∗ ( E ) E /E, ( f/E ) I π y πf/E (1)where f ∗ ( I ) is the internal category in F /f ∗ ( E ) obtained by applying ( f / I ) ∗ ; thisfunctor preserves finite limits, and hence the structure of an internal category. Eachvertical morphism labelled π has inverse image functor sending an object to the‘constant diagram’; as well as a right adjoint π ∗ , this functor always has an E -indexed left adjoint π ! . The functors π ∗ and π ! send an internal diagram to its(internal) limit and colimit, respectively.We call a geometric morphism f : F → E precise (resp. polite ; proper ) ifthe square above satisfies the condition π ! ◦ ( f /E ) I ∗ ( V ) ∼ = ( f /E ) ∗ ◦ π ! ( V ) for everysubterminal object V of [ f ∗ ( I ) , F /f ∗ ( E )]. This can be understood as stating thatthe direct image of f preserves E -indexed (resp. E -indexed inhabited; E -indexeddirected) unions of subobjects.We call a Grothendieck topos supercompact (resp. compact ) if its uniquegeometric morphism to Set is precise (resp. proper), which by Lemma 1.2 occurs ifand only if the terminal object has these properties. The global sections morphism ofa Grothendieck topos is polite if and only if the topos is supercompact or degenerate(that is, the terminal object is either supercompact or initial).Note that Moerdijk and Vermeulen denote π ! by ∞ ∗ because, in the propercase, π ! preserves finite limits and hence is the inverse image functor of a geometricmorphism in the opposite direction. In the precise and polite cases the left adjoint π ! is in general not left exact, so this notation no longer makes sense. Example 1.35.
The presheaf topos [ C op , Set ] is supercompact if and only if C is afunnel in the sense of Definition 1.9. Indeed, if C has a weakly terminal object C then there is by inspection an epimorphism y ( C ) ։ C op , Set ], and converselyif 1 is supercompact then one of the morphisms y ( C ) → C admits at least one morphismto the corresponding object of C .More generally, [ C op , Set ] is compact if and only if C is a multifunnel in the senseof Definition 1.11.An immediate consequence of Definition 1.34 is that we can relativise the con-cepts of supercompactness and compactness to depend on the base topos over whichwe work (so far we have been implicitly working over Set ). Viewing a geomet-ric morphism
F → E as expressing F as a topos over E , an object X of F is E -compact if the composite geometric morphism F /X → F → E is proper, for ex-ample. Conversely, since in this paper we will only be concerned with objects whichare supercompact relative to some fixed base topos S , it makes sense to employ thebroader classes of geometric morphism introduced in [19, Chapter V]. See [16, Corollary B3.2.12] for an explanation of why this square is a pullback. efinition 1.36. Let p : E → S and q : F → S be toposes over S . A geometricmorphism f : F → E over S is S -relatively precise (resp. S -relatively polite ; S -relatively proper ) if its direct image preserves arbitrary S -indexed unions (resp.inhabited S -indexed unions; directed S -indexed unions) of subobjects. Explicitly,this requires that the respective conditions of Definition 1.34 hold for diagrams I in E of the form p ∗ ( I ′ ), where I ′ is a diagram category of the appropriate type in S .We shall assume S is Set in the remainder, and so we drop the ‘ S -’ prefixes.All of these definitions appear hard to work with in general for the simple reasonthat internal diagram categories take a significant amount of computation to expressand work with concretely (which is to say externally) in any given case. However, the S -relative notions conveniently coincide with their external counterparts, in a sensemade precise in Lemma 1.39 below. Also, all of the notions are clearly stable underslicing and composition, from which we can extract general consequences which aresufficient for the purposes of this paper.By inspection, we have the following relationships between Definitions 1.34 and1.36. Lemma 1.37.
Consider a commuting triangle of geometric morphisms:
F E
Set . fq p Then:1. If f is precise, it is relatively precise.2. The morphisms p and q are relatively precise morphism if and only if they areprecise.3. If f is relatively precise and p is precise, then q is precise.The same statements hold when ‘precise’ is replaced by ‘polite’ or ‘proper’. A handy consequence of this for our objects of interest is the following:
Corollary 1.38.
Let f : F → E be a geometric morphism between Grothendiecktoposes. If f is relatively precise (resp. relatively polite, relatively proper), then f ∗ preserves supercompact (resp. ‘supercompact or initial’, compact) objects.Proof. Given an object E of E and a relatively precise (resp. relatively polite,relatively proper) morphism f , consider the triangle: F /f ∗ ( E ) E /E Set . f/E If E is supercompact (resp. ‘supercompact or initial’, compact), then the globalsections morphism of E /E is also precise (resp. polite, proper), so f ∗ ( E ) must besupercompact (resp. supercompact or initial, compact) by Lemma 1.37.3. n order to make more explicit arguments, we now extend Moerdijk and Ver-meulen’s characterisation of relatively proper geometric morphisms in [19, Proposi-tion V.3.7(i)]. Lemma 1.39.
A geometric morphism f : F → E over S is relatively precise(resp. relatively polite, relatively proper) if and only if for any S -indexed (resp. S -inhabited, S -directed) jointly epimorphic family { g i : X i ֒ → f ∗ ( Y ) } of subobjectsin F there exists a jointly epimorphic family { h j : Y j → Y } in E such that each f ∗ ( h j ) factors through some g i .Proof. Suppose f has one of the relative properties and we are given a collectionof subobjects of the relevant type. By assumption, their union is preserved by( f /Y ) ∗ , whence there are subobjects h j : Y j ֒ → Y (the images under ( f /Y ) ∗ of thesubobjects) which are also jointly epic. By construction, each of these must haveimage under f ∗ which factors through one or more of the X i .Conversely, given an S -indexed diagram (of the relevant type) of subterminalobjects { g i : X i → f ∗ ( Y ) } in F /f ∗ ( Y ), let U ֒ → f ∗ ( Y ) be the union of the g i ,and let Y ′ ֒ → Y be its image under ( f /Y ) ∗ in E /Y . Applying f ∗ , we have amonomorphism f ∗ ( Y ′ ) ֒ → U , so we can pull back to get a jointly epimorphic familyof the appropriate shape { g ′ i : X ′ i → f ∗ ( Y ′ ) } over f ∗ ( Y ′ ). This is precisely dataof the required form to apply the hypotheses, and the covering family of Y ′ thusprovided ensures that the union of the images of the g i under ( f /Y ) ∗ is precisely Y ′ . It is an indirect consequence of Lemma 1.39 that the converse of Corollary 1.38cannot hold in general. Indeed, if the only compact object of E is the initial object,such as in Example 3.20 below, then the preservation of compact or supercompactobjects by f ∗ is a vacuous condition, but the required properties in the character-isation of Lemma 1.39 are clearly non-trivial. However, E failing to have enoughsupercompact (resp. compact) objects is the only obstacle. Proposition 1.40.
Let f : F → E be a geometric morphism, and suppose E issupercompactly generated. Then f is relatively precise if and only if f ∗ preservessupercompact objects, and relatively polite if and only if f ∗ preserves ‘supercompactor initial’ objects. If E is merely compactly generated, then f is relatively proper ifand only if f ∗ preserves compact objects.Proof. Given an arbitrary (resp. inhabited) jointly epic collection of subobjects { g i : X i ֒ → f ∗ ( Y ) } in F , consider a (possibly empty) covering of Y by super-compact subobjects h j : Y j ֒ → Y in E . Since each f ∗ ( Y j ) is supercompact (resp.supercompact or initial), pulling back the inclusions g i along f ∗ ( h j ), we concludethat one of the resulting inclusions f ∗ ( Y j ) (if there are any) must be an isomor-phism by Lemma 1.2; in the inhabited case, this is trivially true when f ∗ ( Y j ) isinitial. Hence the f ∗ ( h j ) each factor through one of the g i , whence the criteria ofLemma 1.39 are fulfilled. The compactly generated case, with a directed family ofsubobjects, is analogous.Since the initial object in any topos is strict, the distinction between relativelyprecise and relatively polite morphisms is indeed as small as this proposition makesit seem. Taking S to be Set , S -indexed just means set-indexed, or small. emma 1.41. Given a geometric morphism f : F → E , f ∗ reflects the initial objectif and only if f ∗ preserves it.Proof. If f ∗ reflects 0, then considering the counit f ∗ f ∗ (0) →
0, we conclude that f ∗ f ∗ (0) is initial (by strictness of the initial object), whence f ∗ (0) ∼ = 0 so f ∗ preservesthe initial object. Conversely, if f ∗ preserves 0, then given C with f ∗ ( C ) ∼ = 0 theunit C → f ∗ f ∗ ( C ) ∼ = 0 is a morphism to 0, whence C is initial. Corollary 1.42.
Let f : F → E be a geometric morphism, and suppose E is super-compactly generated. Then f is relatively precise if and only if it is relatively politeand f ∗ reflects the initial object.Proof. A (relatively) precise morphism is (relatively) polite. Considering 0 as anempty union of subobjects of any given object, it is preserved by f ∗ by relativepreciseness, so f ∗ reflects 0 by Lemma 1.41, as required. Note that this implicationholds even when E is not supercompactly generated.Conversely, given that f is relatively polite and f ∗ reflects 0, we have that f ∗ preserves ‘supercompact or initial’ objects by Proposition 1.40, but a supercompactobject X of E is sent to an initial object if and only if it is initial, which is impossible,so f ∗ ( X ) is supercompact, as required. Recall that a geometric morphism f : F → E is a surjection if f ∗ is faithful,or equivalently if f ∗ is a comonadic functor. Meanwhile, a geometric morphism f : F → E is an inclusion (or embedding ) if its direct image f ∗ is full and faithful.See [18, § VII.4] or [16, A4.2] for general results regarding these, which we shallassume familiarity with. In this section we examine how these two types of geometricmorphism interact with supercompact and compact objects, as well as some of theclasses of geometric morphism introduced in the last section.Recall that a geometric inclusion f : F → E is closed if there is some subterminalobject U in E such that f ∗ f ∗ sends an object X to the pushout of the productprojections from X × U . The following result illustrates why the precise, politeand proper morphisms are too restrictive for analysing supercompactly generatedsubtoposes. Lemma 1.43.
An inclusion of toposes f : F → E is proper if and only if it is closed,if and only if it is polite. An inclusion is precise if and only if it is an equivalence.Proof.
The first part is the conclusion of [16, Remark C3.2.9], where it is observedthat a closed inclusion has a direct image functor preserving arbitrary inhabited( E -indexed) unions of subobjects and that closed inclusions are stable under slicing,and conversely that any proper inclusion is closed.Given that f is a closed inclusion, any nontrivial subobject of the correspondingsubterminal object U in E is sent by f ∗ to the initial object in F , so the initialobject is reflected if and only if U is initial, in which case f is an equivalence.The relative versions of these properties are well-behaved with respect to surjec-tions and inclusions. roposition 1.44. Consider a factorisation of a geometric morphism f , F EG . fq p (2)
1. Suppose that p is an inclusion. Then if f is relatively precise (resp. relativelypolite, relatively proper), so is q .2. Suppose that q is a surjection. Then if f is relatively precise (resp. relativelypolite, relatively proper), so is p .It follows that a geometric morphism is relatively precise (resp. relatively polite,relatively proper) if and only if both parts of its surjection-inclusion factorisationare.Proof. We use the characterisation of these properties from Lemma 1.39.1. Given a jointly epic family (resp. inhabited family, directed family) { g i : X i ֒ → q ∗ ( Z ) } in E , we may express Z up to isomorphism as p ∗ ( Y ) (taking Y = p ∗ ( Z ), say),so this can without loss of generality be seen as a family { g i : X i ֒ → f ∗ ( Y ) } . Since f is relatively precise (resp. relatively polite, relatively proper), we have a jointlyepic family { h j : Y j → Y } such that each f ∗ ( h j ) factors through some g i , and hence { p ∗ ( h j ) : p ∗ ( Y j ) → Z } is the required family to fulfill the criterion of Lemma 1.39.2. Given a jointly epic family (resp. inhabited family, directed family) { g i : X i ֒ → p ∗ ( Y ) } in E , we have { q ∗ ( g i ) : q ∗ ( X i ) ֒ → f ∗ ( Y ) } in F being of the desiredform to ensure that there is a covering family { h j : Y j → Y } such that each f ∗ ( h j )factors through some q ∗ ( g i ). But then q ∗ being conservative forces the p ∗ ( h j ) tofactor through g i , as required. Indeed, the intersection of g i with the image of p ∗ ( h j ) is preserved by q ∗ , and one of the sides of the resulting pullback square issent to an isomorphism, which is reflected by q ∗ . Corollary 1.45.
Any topos E with a surjective point is supercompact. Any toposwith a finite jointly surjective collection of points is compact.Proof. The hypotheses correspond to the existence of a surjection
Set → E , or asurjection
Set /K → E with K finite. The unique morphism Set → Set is (anequivalence, and hence) precise, while the morphism
Set /K → Set is proper. Thusby Proposition 1.44.2 and Lemma 1.37, we conclude that E is supercompact (resp.compact) over Set .When the codomain of the geometric morphism is supercompactly (resp. com-pactly) generated, the simpler characterisation of Proposition 1.40 comes to ouraid.
Lemma 1.46.
Suppose that E is supercompactly (resp. compactly) generated and f : F → E is an inclusion. If p is relatively polite (resp. relatively proper), F is alsosupercompactly (resp. compactly) generated.Proof. By Proposition 1.40, the separating collection of supercompact (resp. com-pact) objects in E is mapped by the inverse image of the relatively polite (resp.relatively proper) inclusion f to a separating collection of objects of the same kind(or initial objects) in F . e shall see in Corollary 2.9 that Lemma 1.46 is optimal, in the sense thata topos is supercompactly (resp. compactly) generated if and only if it admits arelatively polite (resp. relatively proper) inclusion into a presheaf topos. Corollary 1.47.
Suppose that E is supercompactly (resp. compactly) generated.Then a geometric morphism f : F → E is relatively polite (resp. relatively proper)if and only if both parts of its surjection-inclusion factorisation have inverse imagespreserving supercompact or initial (resp. compact) objects.Proof.
This follows from applying Proposition 1.40 to the factorisation in Propo-sition 1.44, using Lemma 1.46 to conclude that the intermediate topos must besupercompactly (resp. compactly) generated.More generally, surjections interact well with supercompact and compact objects.
Lemma 1.48.
Suppose f : F → E is a surjective geometric morphism. Then f ∗ reflects supercompact, compact and initial objects.Proof. Since the inverse image functor of f is comonadic, we have an equivalencebetween E and the topos of coalgebras for the comonad on F induced by f . Withoutloss of generality we work with coalgebras.Given a coalgebra ( X, α : X → f ∗ f ∗ ( X )) and a jointly epic family g i : ( U i , β i ) → ( X, α ), since f ∗ preserves arbitrary colimits, the underlying family of morphisms g i : U i → X in F must be jointly epic. Thus if X = f ∗ ( X, α ) is supercompact (resp.compact), one of the g i must be an epimorphism (resp. there is a finite jointlyepic subfamily of the g i ). Since f ∗ moreover creates colimits, we conclude (via thesame colimit diagram employed in Lemma 1.16) that g i is an epimorphism in E too(resp. that the finite subfamily lifts to a jointly epic finite subfamily). Thus ( X, α )is supercompact, as required.Preservation of the initial object by f ∗ , which is equivalent to reflection of 0 by f ∗ by Lemma 1.41, is due to strictness of the initial object forcing f ∗ f ∗ (0) ∼ = 0.It follows from Lemma 1.48 and Corollary 1.42 that a geometric surjection isrelatively precise if and only if it is relatively polite. Recall that a geometric morphism f : F → E is said to be hyperconnected if f ∗ is full and faithful and its image is closed in F under subobjects and quotients (upto isomorphism). See [16, Section A4.6] for some background on hyperconnectedmorphisms, as well as localic morphisms, which shall feature in Section 3, below.When it comes to hyperconnected morphisms into Set , there are various alter-native characterisations; note that these rely on properties of
Set , so are not trueconstructively.
Proposition 1.49.
Let E be a Grothendieck topos. Then the following are equiva-lent:1. The unique geometric morphism E →
Set is hyperconnected.2. E is two-valued : the only subterminal objects are the initial and terminalobjects. . Every object of E is either well-supported (the unique morphism X → is anepimorphism) or initial, but not both.4. E is non-degenerate and has a separating set of well-supported objects.Proof. (1 ⇔
2) The inverse image of a hyperconnected geometric morphism is fulland faithful and closed under subobjects, so in particular the only subobjects of1 in E is 0. Conversely, if E is two-valued, consider the hyperconnected-localicfactorization of the unique geometric morphism E →
Set ; the intermediate toposis the localic reflection of E , equivalent to the topos of sheaves on the locale ofsubterminal objects of E , so is equivalent to Set . Thus the morphism
E →
Set ishyperconnected.(2 ⇔
3) If E is two-valued, the monic part of the epi-mono factorization of X → X ∼ = 0, so X is either initial or well-supported. Conversely,any proper subterminal object fails to be well-supported, so if 3 holds there can beno proper subterminals and E is two-valued.(3 ⇔
4) Since E is a Grothendieck topos, it has some generating set of objects;any such set is still generating after excluding the initial object, and since 0 isdistinct from 1, any generating set contains a non-initial object, so we obtain agenerating set of non-initial objects as required. Conversely, an inhabited colimit ofwell-supported objects is well-supported, so any non-initial object is well-supportedif there is a separating set of well-supported objects.In particular, since supercompact objects are not initial, we obtain a necessaryand sufficient condition for the hypotheses in Section 1.5 to hold: Corollary 1.50. If E is a two-valued (Grothendieck) topos, every supercompactobject in E is well-supported. In particular, a morphism A → B in the subcategory C s of supercompact objects objects of E is an epimorphism if and only if it is epic in E , so every epimorphism in C s is strict. Conversely, if E is supercompactly generated,then C s is closed under cokernels if and only if E is two-valued. Thus we can give the counterexample, promised earlier, to the hypothesis thatepimorphisms in C s are regular when E is two-valued. Example 1.51.
Let M be the free monoid on two generators, viewed as a category,and let E = [ M op , Set ], where the objects are viewed as right M -sets. It is easilychecked that this topos is two-valued, and being a presheaf topos it is supercompactlygenerated. The supercompact objects in this topos are precisely the cyclic right M -sets.Given a cyclic M -set N generated by n , the relation generated by a pair ofmorphisms f, g : N ⇒ M is the one identifying f ( n ) k with g ( n ) k for each k ∈ M .A case-by-case analysis of the possible pairs of elements f ( n ) , g ( n ) demonstratesthat there is no pair of which the epimorphism M ։ E .The main reason for our interest in hyperconnected morphisms, however, is thatthey create the structure of supercompactly (and compactly) generated Grothendiecktoposes which we are studying in this paper. Corollary 1.52. If f : F → E is a hyperconnected geometric morphism betweenGrothendieck toposes, then it is precise. Thus (being surjective) f ∗ preserves andreflects supercompact, compact and initial objects. roof. We extend Moerdijk and Vermeulen’s proof that hyperconnected morphismsare proper in [19, Proposition I.2.4], replacing ∞ ∗ with π ! .Suppose f : F → E is hyperconnected, and consider a diagram of the form[ f ∗ ( I ) , F ] [ I , E ] F E . f I π y πf (3)Since f I is a pullback of f , it is hyperconnected too, so that any V ֒ → f ∗ ( I ) , F ]is of the form ( f I ) ∗ ( U ) for some U ֒ → I , E ] (the restriction of a hyperconnectedmorphism to the subterminal objects is an equivalence). Thus, f ∗ π ! ( V ) = f ∗ π ! ( f I ) ∗ ( U ) = f ∗ f ∗ π ! ( U ) = π ! ( U ) , where the last equality holds since f ∗ is full and faithful. But U = ( f I ) ∗ ( f I ) ∗ ( U ) =( f I ) ∗ ( V ), so f ∗ π ! ( V ) = π ! ( f I ) ∗ ( V ), as required. The same argument applied in slicesgives the result.Preservation of supercompact and compact objects by f ∗ then follows fromProposition 1.40 and Corollary 1.42 (preservation of the initial object is automatic),while reflection follows from Lemma 1.48. Theorem 1.53.
Let f : F → E be a hyperconnected geometric morphism betweenelementary toposes. If F is a Grothendieck topos, then so is E . Assuming this is so,if F is also:(i) is supercompactly generated, or(ii) is compactly generated, or(iii) has enough points, or(iv) is two-valued,then E inherits that property.Proof. Let C be a (full subcategory on a) small separating set of objects in F . Thenevery object of F is a colimit of a diagram in C . Given an object Q of E , we canexpress f ∗ ( Q ) as such a colimit; write λ i : C i → f ∗ ( Q ) with C i ∈ C for the legs ofthe colimit cone. By taking image factorizations of the λ i we obtain an expressionfor f ∗ ( Q ) as a colimit where the legs of the colimit cone are all monomorphisms.Since the image of f ∗ is closed under subobjects, we obtain an expression for f ∗ ( Q )as a colimit of objects of the form f ∗ ( D i ) with D i in E , which moreover are quotientsof objects in the separating subcategory C of F .Thus, since f ∗ creates all small colimits, the quotients of objects in C lying in E form a separating set. Also, E is locally small since F is. Thus by the version ofGiraud’s Theorem that appears in [16, C2.2.8(v)], say, E is a Grothendieck topos.The inheritance of property (i) (resp. (ii)) follows from Corollary 1.52, taking C in the above to be C s (resp. C c ) and noting that the objects f ∗ ( D i ) in the argumentabove are supercompact (resp. compact) in F by Lemma 1.7, whence the D i are soin E by Corollary 1.52.For (iii), if F has enough points, which is to say that there is a collection ofgeometric morphisms Set → F whose inverse images are jointly faithful, then com-posing each point with the hyperconnected morphism to E gives such a collection for . Finally, for (iv), note once again that the restriction of f to subterminal objectsis an equivalence.In spite of our reliance on Corollary 1.52 here, we shall see in Example 3.21 thatwe cannot extend Theorem 1.53(i) or (ii) to relatively precise or relatively propersurjections, although parts (iii) and (iv) do apply in that situation.We can summarise the results from the last two sections as stability results forsupercompactly and compactly generated toposes. Theorem 1.54.
Suppose E is a supercompactly (resp. compactly) generated Grothendiecktopos. If F is:1. The domain of a closed inclusion f : F → E , or more generally, the domainof a relatively polite (resp. relatively proper) inclusion into E ,2. The domain of a local homeomorphism g : F ≃ E /X → E , or3. The codomain of a hyperconnected morphism h : E → F ,then F is also a supercompactly (resp. compactly) generated Grothendieck topos.Proof. Let C s and C c be the (separating) subcategories of supercompact and compactobjects in F , respectively.1. For any inclusion, the images of objects in C s (resp. C c ) under f ∗ wouldform a separating set for E . The stated properties ensure that these objects areall supercompact or initial (resp. compact), so that E is supercompactly (resp.compactly) generated, by Corollary 1.38.2. The objects with domain in C s (resp. C c ) in any slice F /X form a separatingset. These lifted objects inherit the property of being supercompact (resp. compact),by Lemma 1.2 and the standard result ( F /X ) / ( Y → X ) ≃ F /Y .3. This is immediate from Theorem 1.53. So far, we have established properties of ‘canonical’ sites for supercompactly andcompactly generated toposes. In the spirit of Caramello’s work in [4], we obtain inthis section a broader class of sites whose toposes of sheaves have these properties.We will occasionally make use of the following fact regarding representablesheaves, which is easily derived from the fact that the functor ℓ is a dense mor-phism of small-generated sites, in the sense described by Caramello in [8]: Fact 2.1.
A sieve T on ℓ ( C ) in Sh( C , J ) is jointly epimorphic if and only if thesieve { f : D → C | ℓ ( f ) ∈ T } is J -covering. Definition 2.2.
Let C be a small category. A class T of morphisms in C is called stable if it satisfies the following three conditions:1. T contains all identities.2. T is closed under composition. . For any f : C → D in T and any morphism g : B → D in C , there exists acommutative square A BC D, f ′ g ′ gf (4)in C with f ′ ∈ T .These correspond to the necessary and sufficient conditions for T -morphisms to besingleton presieves generating a Grothendieck topology, as presented by Mac Laneand Moerdijk in [18, Exercise III.3]. We call the resulting Grothendieck topologythe principal topology generated by T .In [17], Kondo and Yasuda call a stable class of morphisms semi-localizing , inreference to a related definition in [12]. They call a principal topology an A -topology,presumably because the atomic topology is an example of a principal topology; seeExample 2.21. We have chosen a naming convention that we believe to be moreevocative in this context.Continuing the parallel investigation of compactness, we obtain a related conceptby replacing individual morphisms by finite families of morphisms. Definition 2.3.
Let T ′ be a class of finite families of morphisms with specifiedcommon codomain in C , we say T ′ is stable if1’. T ′ contains the families whose only member is the identity.2’. T ′ is closed under multicomposition, in that if { f i : D i → C | i = 1 , . . . , n } isin T ′ and so are { g j,i : E j → D i | j = 1 , . . . , m i } for each i = 1 , . . . , n , then { f i ◦ g j,i } is also a member of T ′ .3’. For any { f i : D i → C | i = 1 , . . . , n } in T ′ and any morphism g : B → C in C ,there is a T ′ -family { h j : A j → B | j = 1 , . . . , m } such that each g ◦ h j factorsthrough one of the f i .These are necessary and sufficient conditions for T ′ -families to generate a Grothendiecktopology, which we call the topology (finitely) generated by T ′ .Note that we do not require C to have pullbacks in the above definitions, so it issensible to compare them with the usual notion of stability with respect to pullbacks. Lemma 2.4.
Let T be a stable class of morphisms in C with the additional ‘push-forward’ property:4. Given any morphism f of C such that f ◦ g ∈ T for some morphism g of C , wehave f ∈ T .This in particular is true of the class of epimorphisms, for example. Then morphismsin T are stable under any pullbacks which exist in C .Proof. Given a pullback of a T morphism, the comparison between any squareprovided by (4) and this pullback provides a factorization of a T morphism throughthe pullback, so by the assumed property the pullback is also in T , as required. s Kondo and Yasuda remark in [17], we can extend a stable class of morphisms T to the stable class ˆ T of morphisms whose principal sieves contain a member of T (that is, morphisms f such that f ◦ g ∈ T for some g ) without changing theresulting principal topology. Thus we can safely assume that stable classes satisfyaxiom 4 of Lemma 2.4 if we so choose. The advantage of enforcing this assumptionis that it gives a one-to-one correspondence between stable classes of morphisms ina category and the principal Grothendieck topologies on that category, since we canrecover the classes as those morphisms which generate covering sieves. Remark . The class of identity morphisms in any category satisfies axioms 1,2and 3, but in order to satisfy axiom 4 it must be extended to the class of splitepimorphisms, which is easily verified to satisfy all four axioms in any category. Thusthe class of split epimorphisms corresponds to the trivial Grothendieck topology. Itis worth noting also that every split epimorphism is regular and hence strict.We may similarly extend a class T ′ to a maximal class ˆ T ′ . However, for a classof finite families to be maximal, it must be closed under supersets as well as underthe equivalent of the push-forward property of Lemma 2.4, so we have two extraaxioms:4’. Given a finite family f = { f i : D i → C } of morphisms in C such that everymorphism in some T ′ -family over C factors through one of the f i , we have f ∈ T ′ .5’. Any finite family f = { f i : D i → C } of morphisms in C containing a T ′ -family f ′ is also a member of T ′ .The equivalent of the pullback stability statement of Lemma 2.4 is as follows. Scholium 2.6.
Suppose that T ′ is a stable class of finite families of morphisms ina category C satisfying the additional axioms 4’ and 5’. Then T ′ is stable under anypullbacks which exist in C , in that given a family { g i : E i → C } in T ′ and h : C ′ → C such that the pullback of g i along h exists for each i , the family { h ∗ ( g i ) : E ′ i → C ′ } is in T ′ . We leave the proof, and the verification that enforcing axioms 4’ and 5’ givesa one-to-one correspondence between stable classes of finite families and finitelygenerated Grothendieck topologies, to the reader.
Definition 2.7.
Let C be a small category, T a stable class of its morphisms and J T the corresponding Grothendieck topology. We call a site ( C , J T ) constructed inthis way a principal site . Similarly, for a stable class of finite families T ′ on C , wehave a corresponding Grothendieck topology J T ′ ; a site of the form ( C , J T ′ ) shall becalled a finitely generated site . Proposition 2.8.
Let C be a small category and J a Grothendieck topology on it.Then the representable sheaves are all supercompact if and only if J = J T is aprincipal topology for some stable class T of morphisms in C . In particular, thetopos of sheaves on a principal site ( C , J T ) is supercompactly generated.Similarly, the representable sheaves are all compact in Sh( C , J ) if and only if J = J T ′ for a stable class T ′ of finite families of morphisms in C , and the topos ofsheaves on a finitely generated site ( C , J T ′ ) is compactly generated.Proof. By Fact 2.1, for J = J T , given a covering family on ℓ ( C ), the sieve it generatesmust contain the image of a T -morphism; call it g . Then any morphism in the family hrough which g factors generates a covering sieve and so g must be an epimorphism.Thus ℓ ( C ) is supercompact, as required.Conversely, given that ℓ ( C ) is supercompact for every C , let T be the class ofmorphisms f such that ℓ ( f ) is epimorphic. We first claim that T is a stable class.Indeed, axioms 1, 2 and 4 are immediate; to see that axiom 3 holds, suppose that f : C → D is in T and g : B → D is any C morphism. Then we may consider thepullback of ℓ ( f ) against ℓ ( g ) in Sh( C , J ): A ℓ ( B ) ℓ ( C ) ℓ ( D ) . f ′ g ′ y ℓ ( g ) ℓ ( f ) Since A is covered by objects of the form ℓ ( C ′ ), by supercompactness of ℓ ( B ) theremust be an epimorphism ℓ ( C ′ ) ։ ℓ ( B ) factoring through the pullback, and in turnthe sieve it generates must contain an epimorphism in the image of ℓ , so that weultimately recover the square (4) required for axiom 3.Now we show that J = J T . Given a J -covering sieve S on C (generated by afamily of morphisms, for example), the sieve generated by ℓ ( S ) := { ℓ ( g ) | g ∈ S } necessarily covers ℓ ( C ), and therefore ℓ ( S ) contains an epimorphism by supercom-pactness, so that the original sieve must have contained a member of T , which gives J ⊆ J T . Conversely, J T -covering sieves are certainly J -covering, so J T ⊆ J . Thus J is a principal topology, as claimed.The argument for finitely generated sites is almost identical after replacing T -morphisms with finite T ′ -families in the first part, and defining T ′ families to bethose finite families which are mapped by ℓ to jointly epic families in the secondpart.In particular, we may extend a stable class of morphisms T to a stable class offamilies of morphisms T ′ by viewing the morphisms in T as one-element families.Then it is clear that J T = J T ′ .Intermediate between the two classes of sites discussed so far are a class which wecall quasi-principal sites : these are sites ( C , J ) such that for every object C ∈ C ,either the empty sieve is a covering sieve on C or every covering sieve on C containsa principal sieve. Observe that if C ′ is the full subcategory of C on the latter classof objects (which we can always construct over Set ), then Sh( C , J ) ∼ = Sh( C ′ , J T ),where T is the class of morphisms generating principal covering sieves.The following result subsumes Lemma 4.11 of Bridge’s thesis [2]; our work up tothis point allows us to avoid any direct manipulation with sheaves in the proof. Corollary 2.9.
Let ( C , J ) be a small site. Then the inclusion Sh( C , J ) → [ C op , Set ] is relatively precise (resp. relatively proper) if and only if J is a principal (resp.finitely generated) topology. The inclusion is relatively polite if and only if ( C , J ) isa quasi-principal site.Proof. Since [ C op , Set ] is supercompactly generated, by Proposition 1.40, the inclu-sion Sh( C , J ) → [ C op , Set ] is relatively precise if and only if the sheafification functorpreserves supercompact objects; this in particular requires all of the objects ℓ ( C )to be supercompact, which occurs if and only if J is principal by Proposition 2.8.But any other supercompact objects are quotients of representables, so ℓ ( C ) being upercompact for every C is also sufficient. As usual, the relatively proper case isanalogous.For relative politeness, we relax the conditions above to requiring that eachrepresentable is sent to a supercompact or initial object, and note that the emptysieve is covering on C if and only if ℓ ( C ) is initial. In order to better understand the relationship between a principal site and the toposit generates, we employ some results of Caramello in [8], which enable us to describemorphisms in a Grothendieck topos Sh( C , J ) in terms of those in a presenting site( C , J ).For a general site ( C , J ), the functor ℓ : C →
Sh( C , J ) is neither full nor faithful.To describe the full collection of morphisms in the sheaf topos, several notions areintroduced in [8, § h, k : A ⇒ B , we say h and k are J -locally equal (written h ≡ J k ) if there is a J -covering sieve S on A such that h ◦ f = k ◦ f for every f ∈ S .If J is principal (resp. finitely generated) then this is equivalent to saying that thereis some T -morphism which equalizes h and k (resp. a T ′ -family whose membersall equalize h and k ). This leads naturally to the following moderately technicaldefinitions: Definition 2.10.
Let C be a small category and T a stable class of morphisms in C . Then for objects A, B in C , a T -span from A to B is a span CA B, f g (5)such that f is in T . A T -arch is a T -span such that for any h, k : D ⇒ C with f ◦ h = f ◦ k we have g ◦ h ≡ J T g ◦ k .Similarly, for T ′ a stable class of finite families of morphisms on C , a T ′ -span isa finite (possibly empty) family of spans: C i A B, f i g i (6)such that { f , . . . , f n } is in T ′ . A T ′ -multiarch is a T ′ -multispan such that for any h : D → C i , k : D → C i ′ with f i ◦ h = f i ′ ◦ k we have g i ◦ h ≡ J T ′ g i ′ ◦ k .The constituent morphisms in any span or multispan will be refered to as their legs . Lemma 2.11.
Let ( C , J T ) be a principal site. Let Arch T ( A, B ) be the collectionof T -arches from A to B in C . For each T -arch ( t, g ) ∈ Arch T ( A, B ) , there is a(necessarily unique) morphism ℓ ( t, g ) : ℓ ( A ) → ℓ ( B ) in Sh( C , J T ) such that ℓ ( t, g ) ◦ ℓ ( t ) = ℓ ( g ) . The mapping ℓ so defined is a surjection from Arch T ( A, B ) to the setof morphisms from ℓ ( A ) to ℓ ( B ) in Sh( C , J T ) .Similarly, letting mArch T ′ ( A, B ) be the set of T ′ -multiarches from A to B , ℓ induces a surjection from mArch T ′ ( A, B ) to Hom
Sh( C ,J T ′ ) ( A, B ) . roof. This is immediate from [8, Proposition 2.5].Intuitively it seems that the collections
Arch T ( A, B ) “should” be the morphismsof a category. However, while Caramello’s result [8, Proposition 2.5(iv)] suggestsa composition of arches coming from covering families that generate sieves, thiscomposition produces a maximal family of arches presenting the composite ratherthan a single T -arch; there is a similar problem for multiarches. We thereforeexamine what structure exists in general, and identify some sufficient conditionsunder which arches and multiarches admit a composition operation. Lemma 2.12.
Let ( C , J T ) be a principal site. For each pair of objects A and B in C , let Span T ( A, B ) be the collection of T -spans from A to B . Then Span T ( A, B ) admits a canonical categorical structure, where a morphism x : ( t : C → A, g : C → B ) → ( t ′ : C ′ → A, g ′ : C ′ → B ) is a morphism x : C → C ′ with t = t ′ ◦ x and g = g ′ ◦ x . This restricts to give a category structure on Arch T ( A, B ) too.Expanding upon this, if ( C , J T ′ ) is a finitely generated site, there is a canonicalcategorical structure on each collection of T ′ -multispans mSpan T ′ ( A, B ) , where ~x :( t i : C i → A, g i : C i → B ) → ( t ′ j : C ′ j → A, g ′ j : C ′ j → B ) consists of an index j for each index i , and a morphism x i : C i → C ′ j with t i = t ′ j ◦ x i and g i = g ′ j ◦ x i .Note that any permutation of the spans forming a given T ′ -multispan form a T ′ -multispan which is isomorphic in this category. Once again, this structure restrictsto the collections of multiarches. Thus in the best case scenario where it is possible to construct a weak composi-tion on arches, we may obtain a bicategory (see [1, Definition 1.1] for a definition ofbicategory) from the principal site ( C , J T ), whose 0-cells are the objects of C , whose1-cells are T -arches and whose 2-cells are morphisms between these. The relation-ship between this bicategory and the subcategory of Sh( C , J T ) on the representablesis simply that of collapsing the vertical morphisms, in the following sense: Lemma 2.13.
Two T -arches (resp. T ′ -multiarches) from A to B on a princi-pal (resp. finitely generated) site are identified by ℓ if and only if they are in thesame component in the category Arch T ( A, B ) (resp. mArch T ′ ( A, B ) ) described inLemma 2.12.Proof. In one direction, if x : ( t : C → A, g : C → B ) → ( t ′ : C ′ → A, g ′ : C ′ → B ),then by definition the unique morphism ℓ ( t, g ) : ℓ ( A ) → ℓ ( B ) with ℓ ( g ) = ℓ ( t, g ) ◦ ℓ ( t )also satisfies ℓ ( g ′ ) = ℓ ( t, g ) ◦ ℓ ( t ′ ), whence ℓ ( t ′ , g ′ ) = ℓ ( t, g ). Thus ℓ identifies T -spansin the same component.Conversely, applying [8, Proposition 2.5(iii)], two T -arches ( t : C → A, g : C → B ) and ( t ′ : C ′ → A, g ′ : C ′ → B ) induce the same morphism in Sh( C , J T ) if and onlyif there is a T -morphism s : D → A and morphisms h : D → C and h ′ : D ′ → C ′ with t ◦ h = s = t ′ ◦ h ′ and g ◦ h ≡ J T g ′ ◦ h ′ . Expanding on the latter condition, thisimplies the existence of some T -morphism u : E → D equalizing g ◦ h and g ′ ◦ h ′ .But then ( t ◦ h ◦ u, g ◦ h ◦ u ) = ( t ′ ◦ h ′ ◦ u, g ′ ◦ h ′ ◦ u ) is easily shown to be a T -arch,and it admits morphisms h ◦ u and h ′ ◦ u ′ to ( t, g ) and ( t ′ , g ′ ) respectively, whencethese are in the same connected component, as required. Proposition 2.14.
Suppose that T is a stable class of morphisms in a small category C such that axiom 3 of Definition 2.2 provides stability squares weakly functori-ally . That is, calling an ordered pair of morphisms ( h : A → D, s : B → D ) with ∈ T a T -cospan from A to B , suppose the stability axiom defines a mapping from T -cospans to T -spans satisfying the following conditions:1. For any T -morphism t : A → B , the T -span coming from (id A , t ) is isomorphicin Span T ( A, B ) to ( t, id B ) .2. If f : C → D , g : B → D and k : B ′ → B such that f is a T -morphism,the T -span obtained by applying the stability mapping along g and then k isisomorphic in Span T ( B ′ , C ) to that obtained by applying it along g ◦ k .3. If f : C → D , g : B → D and e : C ′ → C such that e and f are T -morphisms,the T -span obtained by applying the stability mapping along f and then e isisomorphic in Span T ( B, C ′ ) to that obtained by applying it along f ◦ e .Then there is a weak composition on T -arches, in the sense that there are mappings ◦ : Arch T ( B, C ) × Arch T ( A, B ) → Arch T ( A, C ) , which are associative and unital up to isomorphism of T -arches. Moreover, thiscomposition is natural in the second component up to isomorphism, in the sensethat for each fixed T -arch ( u, h ) in Arch T ( B, C ) , a morphism x : ( t, g ) → ( t ′ , g ′ ) in Arch T ( A, B ) induces a morphism ( u, h ) ◦ x : ( u, h ) ◦ ( t, g ) → ( u, h ) ◦ ( t ′ , g ′ ) in Arch T ( A, C ) , and the resulting mapping ( u, h ) ◦ − : Arch T ( A, B ) → Arch T ( A, C ) is functorial up to unit and associativity isomorphisms.Proof. Even without the listed conditions, stability provides a putative definition ofthe composition operation: given a consecutive pair of T -arches, we simply applythe stability axiom to the pair of morphisms with common codomain, PC C ′ A B B ′ ; T ∋ t ′′ g ′′ t g t ′ g ′ that the resulting T -span ( t ◦ t ′′ , g ′ ◦ g ′′ ) is a T -arch is easily checked. The extraconditions are needed to make this operation weakly unital and associative. Thenaturality in the second component is a direct consequence of the second condition.For brevity, we leave the analogous statement and proof of Proposition 2.14 forfinitely generated sites to the reader, noting that the analogue of T -cospans will notbe the duals of T ′ -multispans, but the more restrictive shape of diagram relevant tothe stability axiom 3’.One situation where the hypotheses of Proposition 2.14 are satisfied is when C has pullbacks, by Lemma 2.4. Corollary 2.15.
Let ( C , J T ) be a principal site where C has pullbacks, such as thecanonical sites on locally regular categories we shall see in Definition 2.35. Thenthe objects of C , the T -spans on C and the morphisms between these assemble into abicategory. In particular, the composition operations of Proposition 2.14 are naturalin the first component. s usual, the analogous result for finitely generated sites holds, but we omit theproof.Proof. The verification of the conditions in Proposition 2.14 is straightforward; notethat we actually require a specified choice of pullbacks, but the mediating isomor-phisms are provided by universal properties. For the final claim, we observe thatin the third condition of Proposition 2.14, we no longer need to restrict ourselvesto the case where e : C ′ → C is a T -morphism, since we can complete the definingrectangle with a pullback square ( T -morphisms are indicated with double-headedarrows): A ′ A BC ′ C D. y ( f ◦ e ) ′ g ′′ e ′ y f ′ g ′ ge f ◦ e f The morphism e ′ provides the morphism of T -spans corresponding to e to make thecomposition natural (again, up to relevant isomorphisms) in the first component, asclaimed.The commutativity of the associativity and identity coherence diagrams whichare required to formally make this a bicategory are guaranteed by the uniqueness inthe universal property of the pullbacks involved. Remark . The fact that T -arches do not assemble into a bicategory in full gen-erality is not merely an artefact of us having restricted ourselves to the data ofthe stable classes of morphisms (resp. finite families), rather than the principal(resp. finitely generated) Grothendieck topologies they generate. If we expand ourcollections of morphisms to multiarches indexed by arbitrary J T -covering families,Caramello’s construction [8, Proposition 2.5(iv)] does give a canonical family rep-resenting the composite, but it typically fails to be unital, since composing withan identity T -span gives a strictly larger family. We can restrict to J -coveringsieves to avoid this problem, but even then, without pullbacks the composition maynot be weakly associative either, since multi-composition of J -covering sieves is notnecessarily associative in the required sense.Returning to the general case, we observe that we do not need the compositionoperation to be well-defined at the level of T -arches in order to reconstruct the fullsubcategory of Sh( C , J T ) on the representable sheaves. Scholium 2.17.
Let ( C , J T ) be a principal site. Then the full subcategory of Sh( C , J T ) on the representable sheaves is equivalent to the category whose objects are the ob-jects of C and whose morphisms A → B are indexed by the connected components ofthe category Arch T ( A, B ) .Similarly, if ( C , J T ′ ) is a finitely generated site, then the full subcategory of Sh( C , J T ′ ) on the representable sheaves is equivalent to the category whose objectsare the objects of C and whose morphisms A → B are indexed by the connectedcomponents of the category mArch T ′ ( A, B ) .Proof. Observe that in the definition of composition given in the proof of Proposition2.14, any choice of stability square will produce a T -arch (resp. T ′ -multiarch) lying n the same component of Arch T ( A, C ) (resp. mArch T ′ ( A, C )), since this T -arch (resp. T ′ -multiarch) will necessarily be mapped by ℓ to the composite of themorphisms corresponding to the pair of arches (resp. multiarches) being composed.Thus, even without weak functoriality, the composition is well-defined on connectedcomponents, as required.In the subcanonical case, where all T -morphisms are strict epimorphisms (resp.all T ′ -families are jointly strictly epimorphic), the computations from this sectionsimplify greatly. Indeed, ℓ is full and faithful in this case, which means that ev-ery component of each category Arch T ( A, B ) (resp. mArch T ′ ( A, C )) contains aunique (multi)arch of the form (id A , f ). In a T -arch ( t, g ), g coequalizes every pairof morphisms which t coequalizes by definition, whence the morphism ℓ ( t, g ) cor-responds to the unique morphism A → B factorizing g through t ; the morphismsrepresenting multiarches are recovered analogously from the universal properties ofjointly strictly epic families. Rather than directly computing the category of representable sheaves in Sh( C , J T )via T -arches, we might hope to simplify things by first modifying the principal site.In Kondo and Yasuda’s definition of ‘ B -site’, they assume that the underlyingcategory is an ‘ E -category’, which is to say that every morphism is an epimorphism,[17, Definitions 4.1.1, 4.2.1], which seems very restrictive. However, by taking thequotient of C by a canonical congruence, we show here that we may at least assumethat T is contained in the class of epimorphisms of C without loss of generality, sincethe corresponding topos of sheaves is equivalent to that on the original site. Proposition 2.18.
Let ( C , J T ) be a principal site. Then there is a canonical con-gruence ∼ on C such that ( C / ∼ , J T / ∼ ) is a principal site with T / ∼ a subclass of theepimorphisms of C / ∼ , and with Sh( C , J T ) ≃ Sh( C / ∼ , J T / ∼ ) .Similarly, if ( C , J T ′ ) is a finitely generated site, there is a congruence ∼ on C such that ( C / ∼ , J T ′ / ∼ ) is a finitely generated site with T ′ / ∼ a subclass of theepimorphisms of C / ∼ , and with Sh( C , J T ′ ) ≃ Sh( C / ∼ , J T ′ / ∼ ) .Proof. Simply let f ∼ f ′ : C → D whenever there is a morphism h : C ′ → C in T with f h = f ′ h . To verify that this is a congruence, given g ∼ g ′ : D → E equalizedby k : D ′ → D , stability of k along f h = f ′ h gives k ′ ∈ T : C ′′ D ′ C ′ C D E ; k ′ kh ff ′ gg ′ T is closed under composition and gf hk ′ = g ′ f ′ hk ′ , so gf ∼ g ′ f ′ as required.By the definition of the congruence it is immediate that the morphisms in T / ∼ are epimorphisms. The canonical functor ( D , J T ) → ( D / ∼ , J T / ∼ ) is a morphismand comorphism of sites , since it is cover-preserving, cover-lifting and flat by in-spection. In the terminology of [8, Definition 5.14], the quotient functor is J T -full, See Definitions 2.42 and 2.49 below for the definitions of morphisms and comorphisms of sites,respectively. T -faithful and J T / ∼ -dense; indeed, being full and essentially surjective, only the J T -faithfulness needs to be checked, and the definition of the congruence ensuresthat this holds. Thus this morphism of sites induces an equivalence of sheaf toposesby [8, Proposition 7.16].The congruence for a finitely generated site has f ∼ f ′ whenever f h i = f ′ h i foreach h i in a T ′ -family. The remainder of the proof is analogous.The reduced site ( C / ∼ , J T / ∼ ) in Proposition 2.18 can be obtained in variousalternative ways. By construction: Scholium 2.19.
The functor
C → C / ∼ in the proof of Proposition 2.18 is theuniversal functor with domain C sending T -morphisms to epimorphisms (resp. T ′ -families to jointly epimorphic families). As such, it is not surprising that ℓ canonically factors through the functor C →C / ∼ , since ℓ sends T -morphisms to epimorphisms by Fact 2.1. The resulting site alsocoincides with the one obtained by lifting the hyperconnected-localic factorisation ofthe geometric equivalence of toposes to the level of morphisms of sites, as describedin [8, § J T -local equality (resp. J T ′ -local equality) reducesto ordinary equality, so that for example a T -arch from A to B is a T -span as in(5) such that for any h, k : D ⇒ C with f ◦ h = f ◦ k we have g ◦ h = g ◦ k .In particular, by considering the arches in which the T -morphism is an identity,we see that the functor ℓ : ( C / ∼ , J T / ∼ ) → Sh( C , J T ) is faithful , and further that ℓ : ( C , J T ) → Sh( C , J T ) is faithful if and only if every T -morphism is an epimorphism,which is to say that the congruence ∼ is trivial. Corollary 2.20.
Let ( C , J T ) be a principal site and let ∼ be the congruence on C from Proposition 2.18. Then the functor ℓ : ( C , J T ) → Sh( C , J T ) is full if and onlyif every T / ∼ -morphism in C / ∼ is a strict epimorphism.Similarly, if ( C , J T ′ ) is a finitely generated site, then ℓ : ( C , J T ′ ) → Sh( C , J T ′ ) isfull if and only if every T ′ / ∼ -family in C / ∼ is a strictly epimorphic family.Proof. If the hypothesis holds, then the site ( C / ∼ , J T / ∼ ) is subcanonical, so theinduced functor to the topos is full and faithful, whence ℓ with domain ( C , J T ) isfull.Conversely, given that ℓ is full, suppose that t : C → A is in T . Suppose thatwe have g : C → B such that whenever h, k : C ′ → C with t ◦ h = t ◦ k , we have g ◦ h = g ◦ k . Then ( t, g ) is a T -arch, and there is a morphism ℓ ( t, g ) in Sh( C , J T )completing the triangle. By fullness of ℓ , this is the image of a morphism A → B in C / ∼ , and by the definition of ∼ there is at most one such morphism, whence t/ ∼ isa strict epimorphism, as claimed.The argument for finitely generated sites is analogous.This site can alternatively be obtained by considering the lifting of the surjection-inclusion factorisation of the geometric equivalence of toposes to the level of sites,as seen in [8, Theorem 6.3]. Example 2.21.
Recall that a category C satisfies the right Ore condition ifany cospan can be completed to a commutative square. This is exactly the condi-tion needed to make the class of all morphisms of C stable, and the correspondingprincipal topology is more commonly called the atomic topology , J at , while the ite ( C , J at ) is called an atomic site . The above results show that we may reduceany atomic site to one in which every morphism is epic (hence a ‘ B -site’ in theterminology of Kondo and Yasuda). Returning to our study of the subcategories of supercompact and compact objectsfrom the last section, we observe that the epimorphisms they inherit from E alwaysmeet most of the requirements for stability. Lemma 2.22.
For any Grothendieck topos E , let C s , C c the usual subcategories.Then the class T of epimorphisms in C s which are inherited from E satisfies axioms1,2 and 4 for stable classes, while the class T ′ of finite jointly epimorphic familieson C c inherited from E satisfies axioms 1’,2’,4’ and 5’.Proof. Clearly C s and C c inherit identities from E , so T contains these and T ′ contains the singleton families of the identities. Since epimorphisms are stable undercomposition in E , T is closed under composition. Multicomposition of finite jointlycovering families in E is similarly direct, giving the second axiom for T ′ . Axioms 4’and 5’ are also straightforward.By Theorem 1.23, if E is supercompactly generated (resp. compactly generated)then axiom 3 (resp. axiom 3’) must also be satisfied in each case. Note that theconverse fails: stability of E -epimorphisms in C s or stability of E -epimorphic finitefamilies in C c are not sufficient to guarantee that E is supercompactly or compactlygenerated. Indeed, if E a non-degenerate topos where the only compact object isthe initial object, such as that exhibited in Example 3.19, the stability axioms aretrivially satisfied but E is neither supercompactly nor compactly generated.In the remainder of this section, we refine the concepts of principal and finitelygenerated sites in order to obtain a characterisation of the categories C s (resp. C c )of supercompact (resp. compact) objects in supercompactly (resp. compactly) gen-erated toposes. Definition 2.23.
We say a small category C is reductive if it has funneling col-imits and its class of strict epimorphisms is stable. The reductive topology J r on a reductive category is the principal topology generated by its class of strictepimorphisms.We say C is coalescent if it has multifunneling colimits and its class of (strictly)epic finite families is stable. The coalescent topology J c on a coalescent categoryis the finitely generated topology on its class of strictly epic finite families. Remark . Some justification for this naming and notation is warranted.The names of the categories are intended to evoke the presence of funneling (resp.multifunneling) colimits, since any diagram in them of the respective shapes ‘isreduced’ (resp. ‘coalesces’) by composing with a suitable epimorphism (resp. jointlyepimorphic family). Indeed, if we consider a functor from such a category to
Set which preserves strict epimorphisms (resp. jointly strictly epimorphic families), thenthe images of these epimorphisms have the effect of reducing/coalescing equivalenceclasses for the relations generated by the images of these diagrams.The names were also chosen to have their first few letters in common with regular and coherent respectively, since the regular and coherent objects in a topos are espectively subclasses of the supercompact and compact objects. Thus, while the r and c in J r and J c stand for reductive and coalescent respectively, we shall see thatwhen a category is both regular and reductive, the regular topology (see Definition2.37 below) coincides with the reductive topology, so the r on J r could also mean‘regular’; similarly for coalescent and coherent.While not every stable class of finite families need contain a stable class of single-ton morphisms, we record the fact that this does happen when the families involvedare strictly epimorphic families. Lemma 2.25.
Any coalescent category is a reductive category with finite colimitsand a strict initial object.Proof. If C is a coalescent category, it certainly has the required colimits, so itsuffices to show stability of strict epimorphisms. Indeed, if t : D ։ C is strict and g : B → C is any morphism in C , then since { t } is a strictly epic family, there is somestrictly epic family { h j : A j → B | i = 1 , . . . , m } over B such that each g ◦ h j factorsthrough t . Factoring this family through the coproduct A + · · · + A m gives a strictepimorphism completing the required stability square (even in the case m = 0).To see that the initial object is strict, observe that the empty family is a strictjointly epic family on the initial object, so given any morphism A →
0, stability forcesthe empty family to be jointly epic over A , whence A is also an initial object.It would be easy to mistakenly conclude based on the results presented thusfar that the subcategory of supercompact objects in the category of sheaves on areductive category C should be equivalent to C . The flaw in this reasoning lies inthe fact that, while the functor ℓ : C →
Sh( C , J r ) is full and faithful (since ( C , J r )is a subcanoniical site) and this functor preserves strict epimorphisms, it does not preserve all funneling colimits; a similar argument applies for coalescent categories. Example 2.26.
Consider the following categories, C and C ′ on the left and rightrespectively. It is easily checked that they are reductive, with strict epimorphismsidentified with two heads. R R A · B R R AC DB
The coequalization is that suggested by the positioning, so that in the first diagram,the coequalizer of the pair coming from R is the terminal object B , but in thesecond diagram, C is the coequalizer of the pair coming from R .One can calculate directly that the category of supercompact objects in Sh( C , J r ) ≃ Sh( C ′ , J r ) is equivalent to C ′ . Indeed, the functor ℓ : C →
Sh( C , J r ) does not preservethe coequalizer diagram R ⇒ A ։ B .We shall see a further example of a failure of ℓ to preserve coequalizers (andhence funneling colimits) in Example 2.67. In order to understand which colimitsare preserved by ℓ , we apply criteria derived by Caramello in [8, Corollary 2.25],which we recall here; we refer the reader to that monograph for the proof. emma 2.27. Let ( C , J ) be a site, F : D → C a diagram and λ = { λ D : F ( D ) → C | D ∈ D} a cocone under F with vertex C . Then λ is sent by the canonicalfunctor ℓ : C →
Sh( C , J ) to a colimit cone if and only if:(i) For any object C and morphism g : C → C in C , there is a J -covering family { f i : C i → C | i ∈ I } and for each i ∈ I , an object D i of D and an arrow h i : C i → F ( D i ) such that λ D i ◦ h i = g ◦ f i .(ii) For any object C in C and morphisms g : C → F ( D ) , g : C → F ( D ) suchthat λ D ◦ g = λ D ◦ g , there is a J -covering family { f i : C i → C | i ∈ I } such that for each i ∈ I , g ◦ f i and g ◦ f i lie in the same connected componentof ( C i ↓ F ) . Observe that the first condition can be simplified.
Lemma 2.28.
Condition (i) of Lemma 2.27 is equivalent to the requirement that { λ D : F ( D ) → C | D ∈ D} is a J -covering family.Proof. Consider the case where g is the identity on C . There must be some J -covering family { f i : C i → C | i ∈ I } and for each i ∈ I , an object D i of D andan arrow h i : C i → F ( D i ) such that λ D i ◦ h i = g ◦ f i . Since every morphismin this covering family factors through a leg of the colimit cone, the legs of thecone must form a J -covering family. Conversely, given any morphism g : C → C ,since J -covering families are required to be stable, pulling back λ gives the required T -morphism over C to fulfill condition (i).Applying this in the particular case of funneling or multifunneling colimits in aprincipal or finitely generated site ( C , J ), we get the following: Proposition 2.29.
Let ( C , J T ) be a principal site and F : D → C a funnelingdiagram with weakly terminal object F ( D ) and λ = { λ D : F ( D ) → C | D ∈ D} a cocone under F with vertex C . Then λ is sent by the canonical functor ℓ : C →
Sh( C , J T ) to a colimit cone if and only if:(i) λ D ∈ T .(ii) For any object C in C and morphisms g , g : C ⇒ F ( D ) , such that λ D ◦ g = λ D ◦ g , there is a T -morphism t : C ′ → C such that g ◦ t and g ◦ t lie inthe same connected component of ( C ′ ↓ F ) .Similarly, if ( C , J T ′ ) is a finitely generated site and F : D → C a multifunnelingdiagram with weakly terminal objects F ( D ) , . . . , F ( D n ) and λ = { λ D : F ( D ) → C | D ∈ D} a cocone under F with vertex C , then λ is sent by ℓ : C →
Sh( C , J T ′ ) to a colimit cone if and only if:(i) { λ D , . . . , λ D n } ∈ T ′ .(ii) For any object C in C and morphisms g : C → F ( D k ) and g : C → F ( D l ) with ≤ k, l ≤ n , such that λ D k ◦ g = λ D l ◦ g , there is a T ′ -family { t i : C i → C | ≤ i ≤ N } such that g ◦ t i and g ◦ t i lie in the same connected componentof ( C i ↓ F ) .Proof. For (i) in each case, we apply Lemma 2.28 and then the fact that everymorphism in the cone factors through λ D (resp. one of the λ D k ) to deduce, thanksto stability axiom 4 (resp. axioms 4’ and 5’) that condition (i) of Lemma 2.27 isequivalent to the given statement. ondition (ii) in each case is a consequence of condition (ii) in Lemma 2.27,having simply taken the special case D = D = D (resp. D = D k and D = D l ). Conversely, for arbitrary g : C → F ( D ) and g : C → F ( D ) such that λ D ◦ g = λ D ◦ g , we may extend this via any of the morphisms p : D → D and p : D → D in the diagram (resp. p : D → D k and p : D → D l ) sothat λ D ◦ F p ◦ g = λ D ◦ F p ◦ g (resp. λ D k ◦ F p ◦ g = λ D k ◦ F p ◦ g ).Given a T -morphism t : C ′ → C such that F p ◦ g ◦ t and F p ◦ g ◦ t are in thesame connected component of ( C ′ ↓ F ) (resp. a T ′ -family { t i : C i → C } such that F p ◦ g ◦ t i and F p ◦ g ◦ t i are in the same connected component of ( C i ↓ F )), itis clear that g ◦ t and g ◦ t (resp. g ◦ t i and g ◦ t i for each i ) also lie in this samecomponent, as required. Corollary 2.30.
For ( C , J T ′ ) a finitely generated site, a cospan λ : X → Y ← X : λ is mapped by ℓ : C →
Sh( C , J T ′ ) to a coproduct cocone if and only if:(i) { λ , λ } ∈ T ′ ,(ii) Whenever f : C → X and f : C → X have λ ◦ f = λ ◦ f , the emptyfamily is T ′ covering on C , and(iii) Whenever f, f ′ : C ⇒ X are coequalized by λ , there is a T ′ -covering familyon C consisting of morphisms equalizing f and f ′ (and similarly for pairs ofmorphisms into X ).If T ′ -covering families are jointly epic, we can replace the last condition by thecondition that λ and λ must be monic.Proof. This is a direct application of Proposition 2.29 for multifunneling colimits inthe case of a finite discrete diagram, where two morphisms are in the same connectedcomponent of ( C ↓ F ) if and only if they are equal (which is impossible if they havedistinct codomains).Imposing these conditions on all funneling and multifunneling colimit cones, afterusing Lemma 1.12 to decompose multifunneling colimits into finite coproducts andfunneling colimits, we arrive at the following definition. Definition 2.31.
We say a reductive or coalescent category C is effectual if forevery funneling diagram F : D → C with colimit expressed by λ : F ( D ) ։ C , forany object C in C and morphisms g , g : C ⇒ F ( D ) such that λ ◦ g = λ ◦ g ,there is a strict epimorphism t : C ′ ։ C such that g ◦ t and g ◦ t lie in the sameconnected component of ( C ′ ↓ F ).We say a coalescent category is positive if finite coproducts are disjoint andcoproduct inclusions are monomorphisms.With these definitions to hand, we can finally express a definitive correspondenceresult between supercompactly or compactly generated toposes and their canonicalsites from Theorem 1.23. Theorem 2.32.
Up to equivalence, there is a one-to-one correspondence betweensupercompactly generated Grothendieck toposes and essentially small effectual, re-ductive categories. The correspondence sends a topos to its essentially small cate-gory of supercompact objects and a reductive category to the topos of sheaves for thereductive topology on that category.Similarly, there is an up-to-equivalence correspondence between compactly gen-erated Grothendieck toposes and effectual, positive, coalescent categories. roof. Passing from a topos to its subcategory of supercompact (resp. compact)objects and back again gives an equivalent topos by Theorem 1.23; the small categorymust be an effectual reductive (resp. effectual positive coalescent) category, sincethe subcategory of supercompact objects is closed under funneling colimits, so theinclusion must preserve them (and coproducts in a topos are disjoint).In the other direction, since the strict (resp. strict finite family) Grothendiecktopology is subcanonical, a reductive (resp. coalescent) category is included faith-fully as a full subcategory of the corresponding sheaf topos, and all of the rep-resentable sheaves are supercompact (resp. compact). The supercompact (resp.compact) objects are funneling (resp. multifunneling) colimits of the representablesheaves in this topos, but these colimits are preserved by ℓ by construction when thecategory is effectual (resp. effectual and positive), so the category of representablesheaves coincides with the category of supercompact (resp. compact) objects, asrequired. Remark . Note that an effectual and positive coalescent category is also aneffectual reductive category, by Lemma 2.25. However, the corresponding toposesfrom Theorem 2.32 are always distinct, since we have seen that the initial object ofa topos is never supercompact but always compact.We shall extend this correspondence to an equivalence of 2-categories in Section2.6. Since geometric morphisms shall come into play at that point, we add here thefollowing extra definition:
Definition 2.34.
A reductive category C is augmented if it has an initial ob-ject. The augmented reductive topology J r + on such a category has coveringsieves generated by singleton or empty strictly jointly epic families. The resulting augmented reductive site ( C , J r + ) is quasi-principal. As we observed earlier, supercompactness and compactness have been studied in thecontext of regular and coherent toposes, which are toposes of sheaves on regular andcoherent categories respectively, equipped with suitable Grothendieck topologies.Here we recall the definitions of these classes of categories, as well as some moregeneral classes, for comparison with reductive and coalescent categories.
Definition 2.35.
Recall that an epimorphism e in a category C is extremal ifwhenever e = m ◦ g with m a monomorphism, then m is an isomorphism.A category is locally regular if it is closed under connected finite limits, it hasan orthogonal (extremal epi, mono) factorisation system, and every span factorsthrough a jointly monic pair via an extremal epimorphism. Such a category is regular if it also has finite products (equivalently, a terminal object). Clearly, aslice (also called an ‘over-category’) of a locally regular category is regular.We say a category is locally coherent if it is locally regular and finite unions ofsubobjects (including the minimal subobject) exist and are stable under pullback.Such a category is coherent if it also has finite products; independently, such acategory is called positive if it has disjoint finite coproducts.Note that a coherent category may have finite coproducts without these beingdisjoint; see the discussion after Definition 3.9 below. emma 2.36. Every extremal epimorphism in a locally regular category is regular.Proof.
We adapt the proof of Johnstone that in a regular category, covers are reg-ular epimorphisms [16, Proposition A1.3.4]; we omit the composition symbol forconciseness in this proof.Let f : A ։ B be an extremal epimorphism in a locally regular category and let a, b : R ⇒ A be its kernel pair. We show that f coequalizes a and b .Suppose c : A → C has ca = cb , and factorize the span ( f, c ) as an extremalepimorphism followed by a jointly monic pair: BA D C. fcd gh
If we can show that g is monic, then extremality of f will force it to be an isomor-phism, so that c = hg − f factors through f .Given k, l : E ⇒ D with gk = gl , consider the following diagram composed ofpullback squares: P · A · E DA D mn p dk ld
The morphism labelled p is a composite of extremal epimorphisms (by stability)and hence is itself an extremal epimorphism. From this diagram and the precedingassumptions, we have f m = gdm = gkp = glp = gdn = f n , whence ( m, n ) factorsthrough ( a, b ) via some morphism q : P → R , and we have: hkp = hdm = cm = caq = cbq = cn = hdn = hlp, whence hk = hl by epicness of p , but since ( g, h ) was jointly monic, we have k = l ,which completes the proof. Definition 2.37.
The regular topology on a regular or locally regular category issimply the principal topology generated by the extremal epimorphisms, and similarlythe coherent topology on a coherent or locally coherent category is the finitelygenerated topology generated by the finite jointly extremal epic families.By Lemma 2.36, we may replace ‘extremal’ with ‘regular’ in the descriptions ofthe stable classes in this definition, whence we see that these sites are subcanonical.Accordingly we obtain a regular (resp. locally regular, coherent, locally coherent)topos of sheaves on such a site, where here the adjective merely indicates that thetopos can be generated by such a site; any topos is automatically a coherent (andhence regular, locally regular and locally coherent) category . Johnstone explains the reason for the somewhat unfortunate naming convention which we are ex-tending here in [16, D3.3]. y the previous results in this section, any locally regular topos is supercom-pactly generated, and any locally coherent topos is compactly generated. We aretherefore led to wonder when the classes of categories coincide. Theorem 2.38.
A small category is locally regular with funneling colimits if andonly if it is a reductive category with pullbacks.A small category is locally coherent with multifunneling colimits if and only if itis a coalescent category with pullbacks. The two notions of positivity coincide in thiscase.In each case, we can remove the “locally” adjective in exchange for adding aterminal object.Proof.
In one direction, by Lemma 2.36 we have that in a locally regular category,the classes of extremal, strict and regular epimorphisms all coincide, since any strictepimorphism is extremal, and they form a stable class by assumption, whence a lo-cally regular category with funneling colimits is a reductive category with pullbacks.Conversely, given a reductive category C with pullbacks, we must show that C has equalizers, since a category has connected finite limits if and only if it has bothpullbacks and equalizers; the remaining conditions follow from Corollary 1.19 andLemma 1.20, thanks to Lemma 2.4. Given a pair of morphisms h, k : A ⇒ B in C ,consider their coequalizer c : B ։ C . Then C /C is regular, since it has pullbacksand a terminal object (so all finite limits), and it inherits the required factorizationsystem, including that for spans, from C . Therefore there exists an equalizer of h and k as morphisms over C , and it is clear that this will also be their equalizer in C .For the locally coherent case, by considering the strictly epic finite families of subobjects , we see that the fact that unions are stable under pullback ensures thatstrictly epic finite families form a stable class, as required.Conversely, given a coalescent category C with pullbacks, the argument above,Corollary 1.19 and Lemma 1.20 give that C is locally regular. For finite unionsof subobjects, observe that it suffices to consider nullary and binary unions. Theformer are guaranteed by the strict initial object of a coalescent category, seen inLemma 2.25. For the latter, observe that the union can be expressed as the pushout(a multifunneling colimit) along the intersection of the two subobjects (the pullbackof the monomorphisms defining the subobjects), since any subobject containing thegiven pair of subobjects forms a cone under this diagram. The fact that strictly epicfamilies are stable under pullback ensures that these unions are too, again thanksto Lemma 2.4.Finally, the two definitions of positivity consist of (existence and) disjointness offinite coproducts, whence these concepts are equivalent.The reader may have noticed that we did not include the properties of effective-ness for regular and coherent categories in Definition 2.35: Definition 2.39.
A (locally) regular category is effective if all equivalence rela-tions are kernel pairs.We chose a similar name, ‘effectual’, for the concept appearing in Definition 2.31because both effectuality and effectiveness are conditions equivalent to the relevantcategories being recoverable from the associated topos. Indeed, a locally regular, Referred to as
Barr-exactness in older texts; we follow Johnstone in our terminology. ffective category C can be recovered from the topos of sheaves on C for the regulartopology, Sh( C , J r ), as the category of regular objects , which were defined in Example1.5. Similarly, if C is locally coherent, positive and effective, it can be recoveredfrom Sh( C , J c ) as the category of coherent objects , which are defined analogously.As a special case, we recover the familiar correspondences between effective regularcategories and regular toposes, or between effective, positive coherent categories(also known as pretoposes ) and coherent toposes. These results are comparable toTheorem 2.32. The concepts of effectuality and effectiveness are directly related: Proposition 2.40.
Let C be a reductive category with pullbacks. Then if C is effec-tual as a reductive category, it is also effective as a regular category.Proof. First suppose that C is effectual, let a, b : R ⇒ A be an equivalence relationon A , and let λ : A ։ B be its coequalizer. We must show that ( a, b ) is the kernelpair of c . Given g , g : C ⇒ A with λ ◦ g = λ ◦ g , by effectuality of C there is astrict epimorphism t : C ′ ։ C such that g ◦ t and g ◦ t lie in the same connectedcomponent of ( C ′ ↓ F ), where F : D → C is the diagram picking out the parallelpair ( a, b ).The only form a connecting zigzag can have in ( C ′ ↓ F ) is (omitting the mor-phisms from C ′ and any identity morphisms): R R · · ·
RA A A A A, x y x y x y n − x n y n with each x i and y i equal to a or b . By reflexivity of R we may construct a zigzagconsisting of n > g ◦ t, g ◦ t ) through the relation ( a, b ).Clearly if x = y we may omit the first zigzag, and similarly for all of the others,so we may assume that x i = y i . By symmetry of R , if ( x i , y i ) = ( b, a ), this factorizesthrough ( a, b ), which gives an alternative zigzag of the same length in ( C ′ ↓ F ), sowe may assume x i = a and y i = b . If n ≥
2, we may factor through the pullbackof x along y , and transitivity of R means that we get a strictly shorter zigzag;iterating this, we reach a zigzag with n = 1, as required.Finally, taking the (regular epimorphism, relation) factorization of ( g ◦ t, g ◦ t ),we conclude that the resulting relation (and hence ( g , g )) must factor uniquelythrough R , as required.As we shall see in Example 2.68, the converse of Proposition 2.40 fails, which iswhy we did not employ the same name for these concepts. Remark . While we provided a direct proof of Proposition 2.40 for completeness,we could more succinctly have reasoned as follows. When a category is both regularand reductive, the strict and regular topologies on the category coincide. If sucha category is effectual, therefore, all of the supercompact objects in its topos ofsheaves are regular, and hence it must also be effective, since it is equivalent to thecategory of regular objects in its associated topos.More generally, one might take an interest in the regular objects in a super-compactly generated topos, or the coherent objects in a compactly generated topos. owever, this class of objects need not be stable under pullback in general, andhence may not assemble into a locally regular (resp. locally coherent) category.Nonetheless, by considering the induced Grothendieck topology on this subcategory,we obtain a supercompactly generated subtopos of the original topos. Iterating thisprocess recursively, in the countable limit we obtain a maximal pullback-stable classof regular objects, although the resulting subcategory still may not be a locally reg-ular category in the sense of Definition 2.35, since that definition also required thepresence of equalizers. Since it is unclear to us whether this class of objects or thecorresponding subtopos have an interesting universal property, and since we lackinteresting specific examples of this construction, we terminate our analysis here. Morphisms of sites are most easily defined on sites whose underlying category hasfinite limits. However, there is no reason for this property to hold in a generalprincipal or finitely generated site. We must therefore use the more general definitionof morphism of sites, which we quote from Caramello [8, Definition 3.2].
Definition 2.42.
Let ( C , J ) and ( D , K ) be sites. Then a functor F : C → D is a morphism of sites if it satisfies the following conditions:1. F sends every J -covering family in C to a K -covering family in D .2. Every object D of D admits a K -covering family { g i : D i → D | i ∈ I } byobjects D i admitting morphisms h i : D i → F ( C ′ i ) to objects in the image of F .3. For any objects C , C of C and any span ( λ ′ : D → F ( C ) , λ ′ : D → F ( C ))in D , there exists a K -covering family { g i : D i → D | i ∈ I } , a family of spansin C , { ( λ i : C ′ i → C , λ i : C ′ i → C ) | i ∈ I } , and a family of morphisms in D{ h i : D i → F ( C i ) } , such that the following diagram commutes: D i F ( C ′ i ) DF ( C ) F ( C ) h i g i F ( λ i ) F ( λ i ) λ ′ λ ′
4. For any pair of arrows f , f : C ⇒ C in C and any arrow λ ′ : D → F ( C )of D satisfying F ( f ) ◦ λ ′ = F ( f ) ◦ λ ′ , there exist a K -covering family in D{ g i : D i → D | i ∈ I } , and a family of morphisms of C { λ i : C ′ i → C | i ∈ I } , satisfying f ◦ λ i = f ◦ λ i for all i ∈ I and of morphisms of D { h i : D i → F ( C ′ i ) | i ∈ I } , making the following squares commutative: D i DF ( C ′ i ) F ( C ) h i g i gF ( λ i ) emark . It is not difficult to show by induction on finite diagrams that the lastthree conditions are equivalent to the following more condensed condition:Given a finite diagram A : I → C and a cone L ′ over F ◦ A in D with apex D ,there is a K -covering family of morphisms { g i : D i → D | i ∈ I } and cones L i over A in C with apex C ′ i such that L ◦ g i factors through F ( L i ) for each i ∈ I , in thesense that there exist morphisms h i : D i → F ( C i ) with D i DF ( C ′ i ) F ( A ( X j )) h i g i λ ′ j F ( λ ij ) for each i , where λ ′ j and λ ij are the j th legs of cones L ′ and L i respectively.Moreover, when the domain site does have finite limits, these three conditionsreduce to the requirement that F preserves finite limits.A functor is a morphism of sites precisely if ℓ D ◦ F : C →
Sh( D , K ) is a J -continuous flat functor, so that this composite extends along ℓ C to provide theinverse image functor of a geometric morphism Sh( D , K ) → Sh( C , J ). Corollary 2.44.
Suppose ( C , J ) and ( D , K ) are principal (resp. quasi-principal,finitely generated) sites. Then any morphism of sites F : ( C , J ) → ( D , K ) inducesa relatively precise (resp. relatively polite, relatively proper) geometric morphism f : Sh( D , K ) → Sh( C , J ) .Proof. Since the conditions on the sites ensure that the representables are super-compact (resp. supercompact or initial, compact), and the restriction of the inverseimage functor to these is precisely ℓ D ◦ F , we conclude that f ∗ preserves these ob-jects, whence f is relatively precise (resp. relatively polite, relatively proper) byProposition 1.40. Remark . More generally, suppose ( C , J ) is any small-generated site such thatSh( C , J ) is supercompactly (resp. compactly) generated and ( D , K ) is a principal(resp. finitely generated) site. The geometric morphism induced by a morphism ofsites F : ( C , J ) → ( D , K ) has inverse image functor sending any funneling (resp. mul-tifunneling) colimit of representables to a colimit of supercompact (resp. compact)objects of the same shape, whence by Lemma 1.14 it must in particular preservesupercompact (resp. compact) objects. As such, we can replace the hypotheses ofCorollary 2.44 with these weaker conditions if we so choose.Beyond morphisms, it is natural to make the extra step of forming a 2-category ofsites. Indeed, any natural transformation between functors underlying morphismsof sites induces a natural transformation between the inverse images of the cor-responding geometric morphisms; for subcanonical sites, this mapping is full andfaithful. Thus, for example, any equivalence of sites (an equivalence of categorieswhich respects the Grothendieck topologies) lifts to an equivalence between the cor-responding toposes.Having formed these 2-categories of sites, we may examine 2-functors betweenthem. We say that a principal site ( C , J T ) is epimorphic (resp. strictly epimorphic )if T is contained in the class of epimorphisms (resp. strict epimorphisms). An (effec-tual) reductive site is an (effectual) reductive category equipped with its reductive opology. We employ the ad hoc notation of EffRedSite , RedSite , StrEpPSite , EpPSite and
PSite for the 2-categories of effectual reductive sites, reductive sites,strictly epimorphic principal sites, epimorphic principal sites and all principal sitesrespectively, each endowed with morphisms of sites as 1-cells and natural transfor-mations as 2-cells. Clearly, we have forgetful 2-functors:
EffRedSite → RedSite → StrEpPSite → EpPSite → PSite . (7)We apply analogous terminology and notation for the comparable kinds of finitelygenerated sites and coalescent sites. For example, we write EPCoalSite , EffCoalSite , PosCoalSite and
FGSite for the 2-categories of effectual positive coalescent sites,effectual coalescent sites, positive coalescent sites and finitely generated sites, re-spectively. There results an analogous chain of 2-functors:
EffCoalSiteEPCoalSite CoalSite StrEpFGSite EpFGSite FGSite . PosCoalSite (8)Consolidating the results of Section 2.2, we find that several of these forgetfulfunctors have adjoints.
Corollary 2.46.
Let ( C , J T ) be a principal site, ∼ the canonical congruence ofProposition 2.18, ℓ ( C ) the full subcategory of Sh( C , J T ) on the representable sheavesand C s the (essentially small) category of supercompact objects in that topos. Thenthe canonical functors underly morphisms of sites: ( C s ,J r ) ( ℓ ( C ) ,J can | ℓ ( C ) ) ( C / ∼ ,J T / ∼ ) ( C ,J T ) which are the units of reflections to the forgetful functors EffRedSite → StrEpPSite → EpPSite → PSite found in Diagram (7) .Similarly, if ( C , J T ′ ) is a finitely generated site, then with analogous notation,we have morphisms of sites: ( C c ,J c ) ( ℓ ( C ) ,J can | ℓ ( C ) ) ( C / ∼ ,J T ′ / ∼ ) ( C ,J T ′ ) which are units for the reflections of the forgetful functors EPCoalSite → StrEpFGSite → EpFGSite → FGSite appearing in Diagram (8) .All of these units induce equivalences at the level of the associated toposes.Proof.
We omit the straightforward checks that these are indeed morphisms of sites.The universality of the middle and right hand units has been discussed in and eneath Proposition 2.18 and Corollary 2.20; it remains only to show that the finalmorphism ( ℓ ( C ) ,J can | ℓ ( C ) ) → ( C s ,J r ) is universal.Let C ′ be an effectual, reductive category. Then a morphism of sites F : ( C , J T ) → ( C ′ , J r ) corresponds to a geometric morphism Sh( C ′ , J r ) → Sh( C , J T ) whose inverseimage functor restricts to F on the representable sheaves, and so sends these tosupercompact objects in Sh( C ′ , J r ). Since a quotient of a supercompact object issupercompact and inverse image functors preserve quotients, F extends uniquely(up to isomorphism) to a morphism of sites ( C s , J r ) → ( C ′ , J r ) inducing the samegeometric morphism, as required.As usual, the proof for finitely generated sites is analogous.The morphisms appearing in Corollary 2.46 allow us to give another characteri-sation of reductive and coalescent categories. Lemma 2.47.
Let ( C , J T ) be a strictly epimorphic principal site in which T isthe class of all strict epimorphisms of C . Then, assuming the axiom of choice, C is reductive if and only if the (underlying functor of the) composed unit morphism ( C , J T ) → ( C s , J r ) has a left adjoint.Similarly, a strictly epimorphic finitely generated site ( C , J T ′ ) where T ′ consistsof the strict jointly epic families has C coalescent if and only if the morphism of sites ( C , J T ) → ( C c , J c ) has a left adjoint.Proof. If C is reductive and T is its class of strict epimorphisms, consider a super-compact object C in Sh( C , J T ) = Sh( C , J r ). C is a quotient of some representable ℓ ( C ), and so is the colimit of some funneling diagram in ℓ ( C ) with weakly terminalobject ℓ ( C ). Lifting this to a funneling diagram in C , call its colimit L ( C ). Thereis a universal morphism η : C → ℓ ( L ( C )), since the image of the strict epimorphism ℓ ( C ։ L ( C )) forms a cone under the original funnel in Sh( C , J T ). This η is theuniversal morphism from C to a representable object, since given C → ℓ ( D ), wehave that the composite ℓ ( C ) → C → ℓ ( D ) is a morphism in the image of ℓ (since ℓ is full and faithful on a strict principal site) forming a cone under the same funnel,so there is a factoring morphism ℓ ( L ( C )) → ℓ ( D ), as required. This universalitymeans that ℓ ( L ( C )) is well-defined up to isomorphism, and we can use choice toselect a representative for each C ; the universality then ensures that L is functorial,and is a left adjoint to the inclusion C → C s , as required.Conversely, suppose we have a left adjoint functor L : C s → C . Given a funnel in C , consider its colimit in C s ; this is preserved by L , so the colimit exists in C , whichis enough to make C a reductive category.The argument for coalescent categories is analogous, passing via a finite coprod-uct to define L ( C ) in the first part.In other words, any reductive category is a coreflective subcategory of an ef-fectual reductive category, and similarly any coalescent category is a coreflectivesubcategory of an effectual positive coalescent category. Theorem 2.48.
Let SG TOP relprec be the -category of supercompactly generatedGrothendieck toposes, relatively precise geometric morphisms and all geometric trans-formations. Then the object mapping: PSite → SC TOP relprec ( C , J T ) Sh( C , J T ) . xtends to a -functor between these -categories. This functor is faithful on -cellsif we restrict the domain to EpPSite . It is full and faithful on -cells and faithful(up to isomorphism) on -cells if we restrict the domain to StrEpPSite . Finally,it is a -categorical equivalence if we restrict the domain to EffRedSite .Analogously, letting CG TOP relprop be the -category of compactly-generated Grothendiecktoposes and relatively proper geometric morphisms, there is a -functor whose effecton objects is: FGSite → CG TOP relprop ( C , J T ′ ) Sh( C , J T ′ ) . This restricts to an equivalence between CG TOP relprop and the -category EPCoalSite .Finally, there is an equivalence between the -category SG TOP relpol of supercom-pactly generated Grothendieck toposes, relatively polite geometric morphisms and allgeometric transformations, and the -category EffRed + Site of augmented reductivesites of Definition 2.34.Proof.
The claim of 2-functoriality is fulfilled thanks to Corollary 2.44.Faithfulness when we restrict to
EpPSite is by virtue of the observations afterProposition 2.18 that ℓ is faithful (on T -arches and hence) on morphisms comingfrom the site.Fullness on 2-cells and faithfulness on 1-cells when we restrict further to StrEpPSite is more directly derived from full faithfulness of ℓ for such subcanonical sites, and thefact that natural transformations between inverse image functors are determined bytheir components at the representables. Any such natural transformation (includinga natural isomorphism) restricts along ℓ to a natural transformation between theunderlying morphisms of sites.The equivalence when we restrict to EffRedSite is thanks to Theorem 2.32.Indeed, since an effectual reductive category is equivalent to the category of super-compact objects in the corresponding topos, any relatively precise morphism be-tween such toposes restricts to a morphism of sites between the underlying effectivereductive sites.As ever, the argument for finitely generated sites and quasi-principal sites isanalogous.We would be remiss not to mention comorphisms of sites, which in principle should provide an alternative choice for extending the correspondence of Theorem2.32 to a covariant duality.
Definition 2.49.
Given sites ( C , J ) and ( D , K ), a comorphism of sites F :( C , J ) → ( D , K ) is a functor F : C → D which has the cover-lifting property ,that for any object C of C and K -covering sieve S on F ( C ), there exists a J -coveringsieve R on C with F ( R ) ⊆ S .When the topology on the sites is trivial, any functor is a comorphism of sites, andwe can characterise the morphisms induced by comorphisms of sites as the essentialgeometric morphisms. If we restrict to sites on idempotent complete categories,this results in the familiar 2-equivalence between the 2-category Cat ic of smallidempotent complete categories, functors and natural transformations, and the 2-category TOP ess of presheaf toposes, essential geometric morphisms and geometric ransformations, up to reversal of 2-cells: Cat ic ≃ TOP co ess . We see a special case of this in Section 3.1.However, this special case belies the difficulty of characterising geometric mor-phisms coming from comorphisms of sites, even when the sites are subcanonical andtheir Grothendieck topologies have simple descriptions. Without such a characteri-sation, it is difficult to produce a meaningful 2-equivalence statement. We thereforeleave this endeavour as a challenge for the future.
We shall study supercompactly and compactly generated localic toposes in Section3. Even before characterising these, we can make some observations about theirpoints.
Lemma 2.50.
Every point of a localic topos is relatively polite. A localic topos hasa relatively precise point if and only if it is totally connected.Proof.
We use Lemma 1.39. Since
Set is atomic, given a point p : Set → E andan object X of E , it suffices to consider the minimal inhabited covering of p ∗ ( X ) bysingletons { x i : 1 ֒ → p ∗ ( X ) } or, if p ∗ ( X ) is empty, by the identity { ֒ → p ∗ ( X ) } .The characterisation of localic toposes as being generated by subterminal objectsmeans we have a covering of X by subterminals { y j : X j → X } , and since f ∗ ( X j ) isnecessarily subterminal for every j , these factorize through this minimal covering,as required.Since the inverse image functor of a relatively precise point must reflect 0, everynon-initial open U of E must have p ∗ ( U ) = 1, whence the non-initial opens form acompletely prime filter, which is equivalent to E being totally connected.We can say more about the existence of relatively precise points in general. Lemma 2.51.
Let ( C , J ) be a site such that every J -covering sieve is inhabited. Con-sider the object category equipped with its canonical (equivalently, trivial) topology, (1 , J can ) , whose topos of sheaves is equivalent to Set . The unique functor
C → un-derlies a morphism of sites ( C , J ) → (1 , J can ) if and only if C op is filtered, in whichcase the corresponding point of Sh( C , J ) is relatively precise point. Moreover, thedirect image functor of this point is the inverse image functor of the global sectionsmorphism of Sh( C , J ) , so this topos is totally connected.Proof. Checking that the given conditions on J and C are necessary and sufficientto produce the claimed morphism of sites is straightforward: condition 1 is ensuredby inhabitedness of J -covering sieves, while the remaining conditions translate into C op being filtered. Given that C op is filtered, all inhabited sieves on C are necessarilyconnected. It is shown by Johnstone in [16, Example C3.6.17(c)] that if ( C , J ) is acofiltered site in which all J -covering sieves are connected, then Sh( C , J ) is totallyconnected, and in particular locally connected. Since all of the representable objectsare connected (which is to say they are mapped by the left adjoint to the inverseimage functor of the global sections geometric morphism to 1), we see that thiscoincides up to isomorphism with the inverse image functor of the point induced y the morphism of sites, as required. The point Set → Sh( C , J ) therefore hasdirect image preserving all colimits (since it has an extra right adjoint) whence it iscertainly precise and hence relatively precise. Proposition 2.52.
Given a principal site ( C , J T ) , the topos Sh( C , J T ) has a rela-tively precise point if and only if C op is filtered, if and only if Sh( C , J T ) is totallyconnected.Proof. By Theorem 2.48, any relatively precise point of Sh( C , J T ) comes from a mor-phism of sites ( C s , J s ) → (1 , J can ), where C s is the usual subcategory in Sh( C , J T ).Composing with the canonical morphism of sites ( C , J T ) → ( C s , J s ) from Corollary2.46, we conclude that such a morphism exists if and only if one exists ( C , J T ) → (1 , J can ). Applying Lemma 2.51 gives the result.Note that since the conditions on C are independent of the choice of principaltopology it is equipped with, we may deduce that any relatively precise subtopos ofSh( C , J T ) is totally connected when the hypotheses hold. Remark . Clearly any totally connected topos has a unique relatively precisepoint. We do not know of a counterexample of the reverse implication more generally.We immediately recover the following result, which although deducible from [16,Example C3.6.17(c)], does not seem to have been recorded explicitly anywhere thatwe know of.
Corollary 2.54.
Any regular topos is totally connected.Proof.
It suffices to observe that if C has finite limits, C op is filtered, so Proposition2.52 applies.In particular, by [16, Theorem C3.6.16(iv)], the category of models of a reg-ular theory in any Grothendieck topos has a terminal object (easily described asthe model in which every sort is interpreted as the terminal object). The logic ofsupercompactly generated toposes more generally is a subject for a future paper.Recall that Deligne’s theorem states that every (locally) coherent topos hasenough points. However, the original proof of that theorem, [11], involves noth-ing beyond the presentation of such a topos as the topos of sheaves on a finitely-generated site with finite limits. Since the proof only involves taking pullbacks in asingle place, we find that we can easily remove the assumption that the site has finitelimits thanks to the stability axioms of Definition 2.3. For completeness, we do thisextension here, although we must note that this is an almost verbatim reproductionof the version of Deligne’s proof presented by Johnstone in [14, § C , J T ′ ). Given any directed poset P and a functor F : P → C op , we can construct a point of [ C op , Set ] whose inverseimage functor is X colim p ∈ P X ( F ( p )) , since directed colimits commute with finite limits and all colimits. Following John-stone, we call such an F a pseudo-point of C ; we shall write F ′ to denote the inducedinverse image functor described above. emma 2.55. Let F : P → C op be a pseudo-point of C , X a J T ′ -sheaf on C and x = y ∈ F ′ ( X ) . Let f = { f i : V i → V | i = 1 , . . . , n } ∈ T ′ and v ∈ F ′ ( y ( V )) , where y ( V ) is the representable presheaf corresponding to V . Then there exists a refinement G : Q → C op (that is, there exists an inclusion of directed posets i : P → Q with F = G ◦ i ) such that:(a) The images of x and y under the natural map F ′ ( X ) → G ′ ( X ) are distinct,and(b) The image of v in G ′ ( y ( V )) is in S ni =1 im( G ′ ( y ( V i )) → G ′ ( y ( V ))) .Proof. By definition, v corresponds to some morphism F ( p ) → V with p ∈ P . Given p ′ ≥ p , applying stability of T ′ along the composite F ( p ′ ) → F ( p ) → V , we obtain a finite family f ′ p ′ ∈ T ′ on F ( p ′ ) whose members factor through the members of f . Assuch, since X is a J T -sheaf, we have that the morphism X ( F ( p ′ )) → Q n ′ i ′ =1 X ( U p ′ ,i ′ ),where U p ′ ,i ′ are the domains of the morphisms in f ′ p ′ , is a monomorphism.But filtered colimits preserve finite products and monomorphisms, so F ′ ( X ) =colim p ′ ∈ p/P X ( F ( p ′ )) ֒ → Q n ′ i ′ =1 colim p ′ ∈ p/P X ( U p ′ ,i ′ ). In particular, we can findsome index i ′ p ′ such that x, y have distinct images in colim p ′ ∈ p/P X ( U p ′ ,i ′ p ′ ). Nowlet Q be the poset P × { } ⊔ p/P × { } with ordering ( p, m ) ≤ ( p ′ , m ′ ) if and only if p ≤ p ′ and m ≤ m ′ . Then Q is clearly filtered, and we may define the pseudo-point G : Q → C op via the assignment( p, F ( p ); ( p ′ , U p ′ ,i ′ p ′ . Then for any presheaf Y on C , we have G ′ ( Y ) = colim p ′ ∈ p/P Y ( U p ′ ,i ′ ), whence thepseudo-point has the required properties. Lemma 2.56.
Let
F, X, x, y be as in Lemma 2.55. Then there exists a refinement G of F such that:(a) The images of x and y under the natural map F ′ ( X ) → G ′ ( X ) are distinct,and(b) For each object V of C , each T ′ -family f over V and each v ∈ F ′ ( y ( V )) , theimage of v in G ′ ( y ( V )) is in the image of G ′ ( S ( f )) ֒ → G ′ ( y ( V )) , where S ( f ) is the sieve on V generated by f .Proof. Let Z be the set of triples ( f , V, v ), where f ∈ T ′ over the object V and v ∈ F ′ ( y ( V )). Well-ordering Z , we may index its members by some ordinal α . Wenow use transfinite induction to define a sequence of pseudo-points { G β : Q β → C | β ≤ α } as follows: G = F .If β = γ + 1, G β is obtained from G γ by applying the construction of Lemma2.55 to f and the image of v γ in G ′ γ ( V γ ).If β is a limit ordinal, then G β is the unique common refinement of the G γ with γ < β , whose underlying filtered poset is the colimit of the Q γ .It is then easily seen that the images of x and y in G ′ β ( X ) are distinct for all β and that if γ < β then G β is a refinement of G γ such that the image of v γ in G ′ β ( y ( V γ )) is in the image of G ′ β ( S ( f γ )). So G α is the required refinement of F . We assume for the purposes of this proof that stability axiom 3’ provides a choice of T ′ -familycompleting the relevant squares. emma 2.57. Let
F, X, x, y be as in Lemma 2.55. Then there exists a refinement G of F such that the induced point restricts to a point of Sh( C , J T ′ ) and such thatthe images of x and y in G ′ ( X ) are distinct.Proof. Define a sequence of pseudopoints F n by setting F = F and defining F n +1 to be the refinement of F n constructed in Lemma 2.57. Define G to be the colimit ofthe F n ; then we clearly have G ′ ( Y ) = colim n F ′ n ( Y ) for any presheaf Y on C . Hence x and y have distinct images in G ′ ( X ), and if S ( f ) ֒ → y ( V ) is a J T ′ -covering sieve,then each element of G ′ ( y ( V )) derives from an element of F ′ n ( y ( V )) for some n , andhence from an element of F ′ n +1 ( S ( f )). So G ′ defines a point of [ C op , Set ] having thedesired properties which factors through Sh( C , J T ′ ), as required. Theorem 2.58.
Employing the axiom of choice, any compactly generated topos hasenough points.Proof.
Expressing E as a topos of the form Sh( C , J T ′ ) of sheaves on a finitely gen-erated site (always possible, since we may take the canonical site on the compactobjects), we apply Lemma 2.57 to each member of the jointly epimorphic set ofpoints of the presheaf topos [ C op , Set ] to obtain a jointly epimorphic set of pointsof E .Following a comment of Johnstone in [16, Remark D1.5.11], we prove a construc-tive version of this result in the special case that we are given the data required toavoid the axiom of choice. Scholium 2.59.
Suppose that ( C , J T ′ ) is a finitely generated site such that themorphisms of C admit a well-ordering. Then the corresponding toposes have enoughpoints, constructively.Proof. The source of non-constructiveness is the use of the well-ordering principlein the proof of Lemma 2.56. With the given well-orderings on the site, however, wemay construct such a well-ordering directly.First, observe that in the construction of Lemma 2.55, if we have a well-orderingof the elements of P , then the elements of Q , which consist of the set P ×{ }⊔ p/P ×{ } can be endowed with a well-ordering where every element of the former set comesbefore every element of the latter ; this similarly extends to the limit ordinals inthe transfinite induction of Lemma 2.56. Thus we may assume throughout, withoutloss of generality, that P is endowed with such a well-ordering. As such, given apseudopoint F : P → C op , each v ∈ F ′ ( y ( V )) has a canonical minimal representativein the diagram, consisting of the morphism v : F ( p ) → V such that p is the minimalelement in the well-ordering on P carrying a representative, and v is the minimalrepresenting morphism in this hom-set according to the well-ordering on C . Takingthe lexicographic ordering on the minimal representatives ( p, v : F ( p ) → V ), weobtain a well-ordering on F ′ ( y ( V )).We can also well-order the finite families in T ′ using the well-ordering on C ,ordering first by the number of members, and then by the lexicographical orderingon the family’s members.As such, the set Z of triples ( f , V, v ) where f ∈ T ′ has codomain V and v ∈ F ′ ( y ( V )), acquires a well-ordering, as required. Observe that this well-ordering need not be compatible with the partial ordering defined in thatLemma. .8 Examples of Reductive Categories In this subsection and the next section, we present examples of reductive and coales-cent categories, as well as principal and finitely generated sites, in order to addresssome hypotheses about relationships between the concepts presented in this paper.
Example 2.60.
There exist reductive categories without equalizers, products orpullbacks (or even pullbacks of monomorphisms), so which in particular are not lo-cally regular. Indeed, simplifying Example 2.26, consider the category of presheaveson f, g : A ⇒ B . The subcategory of supercompact objects in this topos is simplythe coequalizer diagram A ⇒ B ։ C , so that in particular the pair of monomor-phisms f, g has neither an equalizer nor a pullback, and the product B × B does notexist. We obtain a similar example from any finite category containing a parallelpair of morphisms lacking an equalizer. Example 2.61.
Any discrete category (a category with no non-identity morphisms)with more than one object is a reductive and locally regular category which is notregular.The free finite cocompletion of a discrete category C is a coalescent category, sinceit is equivalent to [ C op , FinSet ], where
FinSet is the category of finite sets , andthis is a coalescent category, whence [ C op , FinSet ] has funneling colimits computedpointwise.
FinSet is in some sense the simplest possible coalescent category, sinceby inspection, Sh(
FinSet , J c ) ≃ Set . Since
FinSet has pullbacks, these free finitecocompletions are coherent categories. Dually, the free finite limit completion of adiscrete category, [ C , FinSet ] op , is a coalescent category with pullbacks. Example 2.62.
We may also consider the category
FinSet + of inhabited finitesets; since a funneling colimit of inhabited finite sets is inhabited and finite, andtaking the pullback in FinSet of an epimorphism in
FinSet + along a morphismwith inhabited domain gives another epimorphism with inhabited domain, we havethat FinSet + is another example of a reductive category without pullbacks (sincethe two inclusions 1 ⇒ FinSet having an equivalent topos of sheaves; since the coalescent topology on
FinSet is arefinement of the augmented reductive one, we see that we have a (relatively proper)inclusion of toposes:
Set ≃ Sh(
FinSet , J c ) ֒ → Sh(
FinSet + , J r ) , which is not an equivalence since the sheaves represented by the finite sets of car-dinality at least 2 are supercompact in the latter topos but merely compact in theformer. Example 2.63.
Expanding on the dual construction, we have that Sh(
FinSet op , J c )embeds into the classifying topos for the theory of objects , [ FinSet , Set ]. We men-tion this as a counterexample to an extension of Lemma 2.50 to the idea that apoint of any Grothendieck topos is polite: the points of [
FinSet , Set ] correspondto sets (objects of
Set ), and the correspondence sends a geometric morphism to the This expression for the free finite cocompletion applies if and only if C has finite hom-sets, since thisis necessary and sufficient for the representable presheaves to lie in [ C op , FinSet ]. This is trivially thecase when C is discrete. et which is the image of the representable functor y (1). But y (1) is supercompact,so a point corresponding to any set with more than one element fails to be polite. Example 2.64.
For yet another related example, consider the simplex category ∆,whose objects are inhabited finite ordinals,[ n ] = { , . . . , n − } , n ≥ ⇒ [2] wouldbe the empty ordinal which is not an object of ∆).∆ has funneling colimits: given any collection of morphisms into the object [ n ]of ∆, their colimit is the quotient of [ n ] identifying f ( x ) , f ( x ) + 1 , · · · , g ( x ) (or g ( x ) , g ( x ) + 1 , · · · , f ( x )) for each parallel pair f, g : [ n ′ ] ⇒ [ n ] in the diagram andeach x ∈ [ n ′ ]. Moreover, each epimorphism g : [ n ] ։ [ m ] is split (so in particular isstrict) by the monomorphism min( g − ) : [ m ] ⇒ [ n ], say.In particular, by Remark 2.5 the collection of strict epimorphisms is stable, whichmakes ∆ a reductive category with Sh(∆ , J r ) = [∆ op , Set ], the topos of simplicialsets . While this tells us little about the theory of strict linear intervals classified by[∆ op , Set ] (referred to as ‘orders’ in [18][ § Example 2.65.
To contrast Examples 2.61 and 2.64, we recall an example of asupercompactly generated topos which is not equivalent to a presheaf topos; thisshould be contrasted with Proposition 3.1, below.Consider the Schanuel topos, Sh(
FinSet opmono , J at ). We see that ( FinSet opmono , J at )is an atomic site with pullbacks, but moreover it is a reductive and regular site, sinceall of the morphisms in the category are regular epimorphisms which are stable underpullback.We know of two ways to show that this topos is not a presheaf topos. The firstis to show that the site is effectual, which explicitly involves providing an algorithmwhich, given a cofunnel F in FinSet mono with weakly initial object F ( D ) and x, y : F ( D ) ⇒ C equalized by its limit, constructs an inclusion i : C ֒ → C ′ and aconnecting zigzag between i ◦ x and i ◦ y in ( F ↓ C ′ ).It follows that the supercompact objects (equivalently, atoms) in the Schanueltopos are precisely the representable sheaves coming from FinSet opmono . However, if I is a finite set, A is any set of cardinality larger than that of I , and we have inclusionsfrom a further finite set B into both A and I , there can be no monomorphismcompleting the triangle, AI B, whence I is not injective in FinSet mono and hence is not projective in
FinSet opmono .It follows that no object of
FinSet opmono is projective, whence the Schanuel toposhas no indecomposable projective objects, but is non-degenerate, and so is not apresheaf topos (the category of representable presheaves in a presheaf topos beingrecoverable up to idempotent completion as the projective indecomposable objects,which in particular must be supercompact). n alternative proof, which we thank Olivia Caramello for describing to us, is toobserve that the category of representables in a presheaf topos can be identified, upto idempotent-completion, with the full subcategory of finitely presentable objectsin its category of points. We also have that a presheaf topos is atomic if and onlyif the representing category is a groupoid. The Schanuel topos classifies infinitedecidable objects, so its category of points corresponds to the category of infinitesets (and monomorphisms); since any inhabited full subcategory of this categoryis not a groupoid (every infinite set has an injective endomorphism which is notinvertible) and the topos is non-degenerate, it again follows that this cannot be apresheaf topos. Example 2.66.
Since any abelian category is effective regular, any small abeliancategory with funneling colimits is reductive. This is the case for the finitely pre-sented (right) modules of a (right) Noetherian ring, say, since these coincide withfinitely generated modules and so is closed under quotients in the large categoryof modules. For example, the category of finitely generated abelian groups is areductive category with finite limits and colimits.In order to construct a small abelian category which does not have funnelingcolimits, we look for a coherent ring R whose collection of finitely generated idealsis not closed under infinite intersections (note that if the ring is an integral domain,coherence ensures that it will be closed under finite intersections). Let I be aninfinitely generated ideal which is obtained as such an intersection. In the largecategory of modules, we may identify R/I as the colimit of the funneling diagramconsisting of the inclusions of finitely generated sub-ideals of I , along with theparallel zero maps, into the ring, viewed as a (right) module over itself. If thecolimit of this diagram existed in the category of finitely presented R -modules, itwould have to be a quotient of R by some finitely generated ideal, but by constructionthere is no initial finitely presented quotient under this diagram, whence the colimitdoes not exist.Consider the ring R of eventually constant sequences valued in the field on twoelements (with point-wise operations). Observe that all finitely generated ideals inthis ring are principal. An ideal generated by a sequence g is the cokernel of themodule homomorphism R → R sending x to x · (1 − g ), so this is indeed a coherentring. For each index i we have a ‘basis element’ e i which is 1 at i and 0 elsewhere.The ideal I i generated by (1 − e i ) consists of those sequences which are 0 at i .Consider T ∞ j =1 I j : this consist of sequences which are non-zero only at odd indices,but by the eventually constant criterion, no single sequence can generate this ideal,whence the ideal fails to be finitely generated.We would like to thank Jens Hemelaer for helping us to identify the sufficientstructure needed to find this counterexample and Ryan C. Schwiebert for identifyinga ring realising that structure (via math.stackexchange.com).Note that a non-trivial abelian category cannot be coherent or coalescent sincethe initial object is not strict in such a category (cf Lemma 2.25). Example 2.67.
As an example of a regular and reductive category that fails to beeffective, let alone effectual, we adapt an example of Johnstone, [16, before ExampleA1.3.7].Consider the category TF fg of finitely-generated torsion-free abelian groups. Byconsidering it as a reflective subcategory of the category of finitely generated abeliangroups, we find that it is regular and has all coequalizers. We can moreover check hat it has funneling colimits, since the full category of abelian groups is cocompleteand any quotient of a finitely generated group is finitely generated (to obtain thequotient in TF fg , the torsion parts of such a quotient are annihilated). Thus itis a reductive category with finite limits. However, as Johnstone points out, theequivalence relation R = { ( a, b ) ∈ Z × Z | a ≡ b mod 2 } ∼ = Z × Z is not a kernel pair of any morphism in TF fg , so this category is not effective regular. Example 2.68.
For an example of a regular and reductive category that is effectivebut not effectual, we modify another of Johnstone’s examples, [16, Example D3.3.9].Let
Set ω be the full subcategory of Set on the finite and countable sets. This haspullbacks and funneling colimits which are stable under pullback, inherited from
Set . Moreover, all epimorphisms are regular, so this is a regular and reductivecategory (indeed, it is a coherent and coalescent category too!). It also inherits theproperty of being effective from
Set .However, it is not effectual. Johnstone exhibits the following coequalizer dia-gram:
N N , id s (9)where s is the successor function. He concludes by considering the natural numberobject in Sh( Set ω , J c ) that this coequalizer is not preserved by the canonical functor ℓ : Set ω → Sh(
Set ω , J c ); we could deduce the failure of effectuality from that.Instead, we prove it directly by considering the morphisms N N , id g :=2 ×− which are clearly coequalized by the epimorphism in (9). If Set ω were effectual,there would exist some epimorphism t : X ։ N such that t and g ◦ t lie in the sameconnected component of ( X ↓ F ), where F is the parallel arrow diagram (id , s )whose coequalizer is shown in (9). However, given any finite zigzag, X X X X X X N · · · NN N N N , t g ◦ tx y x y n − x n y n where x i and y i are id or s , composing with any morphism m : 1 → X such that t ◦ m > n as elements of N , we conclude that since the difference between the image of m in the first copy of N and the last copy of N is at most n , we have a contradiction.That is, no zigzag of finite length is sufficient to connect t and g ◦ t in ( X ↓ F ). A natural question is how the classes of supercompactly or compactly generatedtoposes interact with the most well-studied class of toposes, the localic toposes. Re- all that a Grothendieck topos is localic if it is of the form Sh( L ), the categoryof (set-valued) sheaves on some locale L . These toposes can equivalently be char-acterised by the fact that their set of subterminal objects is separating, and thesesubterminals form a frame isomorphic to the frame of opens of the locale. As usual, the supercompactly generated case of localic toposes are the more straight-forward case to present. A 1-categorical account of the dualities in this section canbe found in the work of Caramello on Stone-type dualities [3, § Proposition 3.1.
A localic topos is supercompactly generated if and only if it isequivalent to [ C op , Set ] for some poset C . Moreover, any poset is an instance of areductive category.Proof. If E is supercompactly generated, each subterminal must be covered by itssupercompact subobjects, so the supercompact subterminal objects generate thetopos.A subterminal object is supercompact if and only if it has no proper cover bystrictly smaller subterminals. This forces the canonical topology on the supercom-pact objects to be trivial, whence E ≃ [ C op s , Set ]. By considering the expression of E as a category of sheaves on a locale, we see that the supercompact objects mustall be quotients of subterminals, and hence themselves subterminals, so E really isthe category of presheaves on C s . Definition 3.2.
We say a locale is an
Alexandroff locale if it has a sub-base of su-percompact opens. These are precisely the locales presenting the toposes appearingin Proposition 3.1.There are several ways to extend this to a full duality, of which we presenttwo. The first is to consider all order-preserving functions between the posets, orequivalently all functors between them when they are viewed as categories. At thelevel of toposes, these correspond to essential geometric morphisms, and descendingback to the level of Alexandroff locales they correspond to locale maps whose inverseimage mappings preserve arbitrary intersections of opens, which we call completelycontinuous maps.We may also regard the category of posets as a 2-category (or more precisely a(1 , , roposition 3.3. Let
Pos be the -category of posets and order-preserving func-tions with their pointwise ordering. Let LocPSh ess be the -category of localicpresheaf toposes and essential geometric morphisms with geometric transformations.Let AlexLoc cc be the -category of Alexandroff locales and completely continuousmaps between these with the pointwise ordering on frame homomorphisms. Then wehave equivalences of -categories: Pos co ≃ LocPSh ess ≃ AlexLoc cc . As Caramello remarks in [3], there are two canonical ways to recover a topo-logical duality from this localic result (we omit the extension of these dualities topreorders here). Since the toposes of sheaves on Alexandroff locales are presheaftoposes, Alexandroff locales always have enough points; each point correspondsto an inhabited, upward-closed subset of the poset. The corresponding presheaftoposes moreover have enough essential points, and (again since posets are triviallyidempotent-complete) these correspond to elements of the posets.Classically, we have a correspondence between preorders and
Alexandroff spaces ,which are characterised by the fact that their open sets are closed under arbitraryintersections. The correspondence sends a preorder to its underlying set, topologizedby making subsets which are upward-closed with respect to the ordering open, andconversely it puts the specialization order on the points of an Alexandroff space;we recognize this as the space of essential points of the associated presheaf toposon the opposite of the preorder. On morphisms, there is a correspondence betweenorder-preserving functions and all continuous maps. Indeed, we have that:
Lemma 3.4.
Any continuous map between Alexandroff spaces is completely contin-uous (the inverse image mapping preserves arbitrary meets of open sets).Proof.
For a map of topological spaces, the inverse image map is defined on allsubsets of points of the codomain, and preserves arbitrary intersections of these.Combined with the fact that an arbitrary intersection of opens is open, it followsthat the inverse image map preserves arbitrary intersections of opens.The disparity between the correspondence in Proposition 3.3 and the classicalduality is non-trivial: there are continuous maps between Alexandroff locales whichare not completely continuous.
Example 3.5.
Consider the ordinals ω and ω + 1 as posets . The correspondingAlexandroff spaces X ω and X ω +1 have frames of opens ( ω + 1) op and ( ω + 2) op ,respectively, since any upward-closed subset is either empty or has a least element.These are, of course, also the frames of opens of Alexandroff locales, being the framesof subterminals in [ ω, Set ] and [( ω + 1) , Set ], respectively.The collection of all points in these locales correspond to the inhabited, upward-closed subsets of ω op and ( ω + 1) op , of which there are ω + 1 and ω + 2 respectively.Of these, all but one of the points are essential: we have an essential point for eachelement of the original poset, plus a limiting non-essential point corresponding tothe upward-closed subset with no least element.The topological spaces X + ω and X + ω +1 obtained by topologizing the full collectionsof points are the respective sobrifications of X ω and X ω +1 . In each case, the added As is conventional, we identify the elements of an ordinal α with the ordinals β < α . oint (indexed by the limit ordinal ω ) is contained in every open set indexed by n < ω , but the singleton consisting of that point is not open. That is, the opensets in X + ω and X + ω +1 are not closed under arbitrary intersection, so these are notAlexandroff spaces!Consider the frame homomorphism ( ω + 1) op → ( ω + 2) op defined by ω ω + 1and n n for n < ω . This necessarily corresponds (contravariantly) to a continuousmap X + ω +1 → X + ω ; indeed, it is the map sending the points ω and ω + 1 to ω andsending n n for n < ω . However, it fails to correspond to any continous map X ω +1 → X ω , since it does not preserve the intersection of the family of opensindexed by n < ω .In summary, we have: Corollary 3.6.
Let
AlexSp be the -category of Alexandroff spaces and continuousmaps with the pointwise specialization ordering. The equivalences of Proposition 3.3extend to: Pos co ≃ LocPSh ess ≃ AlexLoc cc ≃ AlexSp co . The second way to extend the correspondence between posets and locales toa duality is to allow all locale maps whose inverse images preserve supercompactopens, which by Theorem 2.48 correspond to morphisms of sites between posetsequipped with their trivial topologies; equivalently, they corresponds to the flatfunctors between these posets.
Definition 3.7.
Let C , D be posets and let f : C → D be an order-preservingfunction. We say f is flat if: • For any d ∈ D there exists c ∈ C such that d ≤ f ( c ); • For any element d ∈ D and any elements c, c ′ ∈ C such that d ≤ f ( c ) and d ≤ f ( c ′ ) there exists c ′′ ∈ C such that c ′′ ≤ c , c ′′ ≤ c ′ , and d ≤ f ( c ′′ ). Proposition 3.8.
Let
Pos f be the -category of posets and flat functions with theirpointwise ordering. Let LocPSh rpre be the -category of localic presheaf toposesand relatively precise geometric morphisms with geometric transformations. Let AlexLoc sc be the -category of Alexandroff locales and locale maps between thesewhose inverse images preserve supercompact opens, with the pointwise ordering onframe homomorphisms. Then we have equivalences of -categories: Pos op f ≃ LocPSh rpre ≃ AlexLoc sc . Definition 3.9.
In this paper, a join semi-lattice is a poset having all finite joins,including the bottom element. We say a join semi-lattice is distributive if for anytriple of objects ( a, b, c ) with a ≤ b ∨ c , there are elements b ′ ≤ b and c ′ ≤ c suchthat a = b ′ ∨ c ′ ; see [13, § II.5]. This in particular holds in any distributive lattice.Distributivity, which inductively extends to arbitrary finite joins, is precisely thecondition ensuring that the collection of finite join covers is a stable class of finitefamilies. Note that finite join covers are coproduct injections, so that any distributivejoin semi-lattice is a coalescent category . Moreover, a distributive join semi-latticehas pullbacks if and only if it is a distributive lattice, so any distributive lattice isan example of a coalescent and coherent category which fails to be positive. ven if we had not insisted on the presence of a bottom element in a join semi-lattice, by taking c = a in the definition of distributivity, we would have that anypair of elements in a distributive join semi-lattice has a lower bound, although thereneed not be a greatest such. Lemma 3.10.
Suppose that a, b, c are elements of a distributive join semilattice. Ifthe meets a ∧ b and a ∧ c exist then so does a ∧ ( b ∨ c ) , and the distributive law holds: a ∧ ( b ∨ c ) = ( a ∧ b ) ∨ ( a ∧ c ) . Similarly, if b ∧ c exists then so does ( a ∨ b ) ∧ ( a ∨ c ) via the dual distributivity law: a ∨ ( b ∧ c ) = ( a ∨ b ) ∧ ( a ∨ c ) . Proof.
Suppose a ∧ b and a ∧ c exist. Certainly ( a ∧ b ) ∨ ( a ∧ c ) is a lower bound forboth a and b ∨ c . Given any element x with x ≤ a and x ≤ b ∨ c , by assumptionthere exist b ′ ≤ b , c ′ ≤ c with b ′ ∨ c ′ = x ≤ a . Thus b ′ ≤ a ∧ b and c ′ ≤ a ∧ c , whence x = b ′ ∨ c ′ ≤ ( a ∧ b ) ∨ ( a ∧ c ), as required.Now suppose b ∧ c exists. This time, a ∨ ( b ∧ c ) is a lower bound for both a ∨ b and a ∨ c . Given y such that y ≤ a ∨ b and y ≤ a ∨ c , firstly we have a ′ ≤ a and b ′ ≤ b with a ′ ∨ b ′ = y . Then b ′ ≤ y ≤ a ∨ c so we have a ′′ ≤ a and c ′ ≤ c with a ′′ ∨ c ′ = b ′ . In summary, y = a ′ ∨ a ′′ ∨ c ′ ≤ a ∨ c ′ ≤ a ∨ ( b ∧ c ) , since c ′ ≤ c and c ′ ≤ b ′ ≤ b . Remark . This result looks like it should be deducible from Scholium 2.6, sincewe can view meets as pullbacks. That result does allow us to prove that the dis-tributivity laws hold when all of the terms are well-defined, but it doesn’t seem toprovide enough constraints to construct one meet from others as we have done inthe proof above.
Proposition 3.12.
A localic topos is compactly generated if and only if it is thecategory of sheaves on a distributive join semilattice with respect to the topology thatmakes finite joins covering.Proof.
Unlike in the supercompact case, the compact subterminal objects no longerpopulate all of C c , but a similar argument applies: if C is the subcategory of E onthe compact subterminal objects, then C is a distributive join semilattice since afinite union of compact subterminal objects is compact, and so E = Sh( C , J ) where J is the topology on C whose covering families are finite joins.By considering the locale of subterminal objects in the topos of sheaves on adistributive join semi-lattice, we recover a constructive version of the topologicalduality results for distributive join semi-lattices presented by Gr¨atzer in [13, § II.5] .Without meets, one cannot define prime ideals directly as one would in a distributivelattice. However, one can define prime filters (or prime dual ideals as they are calledin [13]) and define prime ideals as complements of these. Working constructively, itmakes more sense to work with the prime filters, especially given that the topology Note that Gr¨atzer uses the term
Stone space to refer to the result of the general dualising constructionpresented there, rather than the more specific meaning of this term. hat Gr¨atzer defines on the prime ideals has a sub-base consisting of prime idealswhich do not contain a particular element of the distributive semilattice. Indeed, thepoints of the locale corresponding to (the localic topos of sheaves on) a distributivejoin semilattice, when viewed as completely prime filters in the frame of opens,correspond bijectively by restriction to prime filters in the distributive join semi-lattice. When there are enough prime filters, which is always the case if we allowourselves to assume the axiom of choice, the subsets of the corresponding soberspace of filters containing a given element of the semilattice do form a sub-base forthe topology, too.In [13, Theorem II.5.8], Gr¨atzer characterises the spaces arising from join semi-lattices via this duality as the T spaces X having a base of compact open subsetsand satisfying the additional condition,(S2) If F is a closed set in X , { U k | k ∈ K } is a directed family of compactopens in O ( X ) op , and U k ∩ F = ∅ for all k , then (cid:0)T k ∈ K U k (cid:1) ∩ F = ∅ .Gr¨atzer himself describes this condition as “complicated”. It follows from our rea-soning above that we may discard (S2) by replacing the T condition with thecondition that X should be sober: Corollary 3.13.
A sober topological space is the space of prime ideals of a distribu-tive join semilattice if and only if it has a sub-base of compact open sets. Moreover,the space is compact if and only if the semilattice has a top element, and is coherentif and only if the distributive semilattice is a distributive lattice.
Celani and Calomino also arrive at this simplification in [9, Theorem 20] , al-though their direct proof is that sobriety of a T space is equivalent to condition (S2)takes over a page. Celani also extended the correspondence between semilattices andspaces of Gr¨atzer to a duality between meet semi-lattices and a class of ordered topo-logical spaces, which on the space side required introducing meet relations betweenspaces, in [10, Definition 22], which do not in general reduce to continuous maps.We shall not explicitly reproduce that duality result here, although we observe thatit might be possible to recover it by considering comorphisms of join semilatticesites.Our topos-theoretic approach gives us an alternative duality result analogous toProposition 3.8. Translating the definition of morphism of sites into this setting, wearrive at the following definition. Definition 3.14.
Let C , D be distributive join semilattices. An order-preservingmap f : C → D is a distributive join homomorphism if it: • Preserves finite joins (including the bottom element). • For any d ∈ D there exists c ∈ C such that d ≤ f ( c ); • For any element d ∈ D and any elements c, c ′ ∈ C such that d ≤ f ( c ) and d ≤ f ( c ′ ) there exists c , . . . , c n ∈ C and d , . . . , d n ∈ D such that c i ≤ c , c i ≤ c ′ , d i ≤ f ( c i ) and d = d ∨ · · · ∨ d n . Corollary 3.15 (Stone duality for distributive join-semilattices) . Let
DJSL f bethe -category of distributive join semilattices, distributive join homomorphisms, Note that Celani et al work dually with distributive meet semilattices, but this does not affect theresults to any significant degree nd pointwise comparison morphisms. Let LocCG rpro be the -category of com-pactly generated localic toposes, relatively proper geometric morphisms, and geomet-ric transformations. Let preCoh c be the -category of locales having a sub-base ofcompact opens, continuous maps whose inverse image mappings preserve compactopen sets, and pointwise comparisons of these. Then we have equivalences: DJSL f ≃ LocCG rpro ≃ preCoh c Given the axiom of choice, by Theorem 2.58, we may identify the objects of the lattercategory with the sober topological spaces having a sub-base of compact open subsets.
Observe that this duality is not merely an extension of the Stone duality be-tween distributive lattices and coherent locales recalled by Caramello in [3, § By applying Theorem 1.54, we can immediately conclude that the above character-isations apply to the localic reflections of supercompactly and compactly generatedtoposes.
Corollary 3.16.
The localic reflection of any supercompactly generated topos is a(localic) presheaf topos. The localic reflection of a compactly generated topos is atopos of sheaves on a distributive join semilattice.
Example 3.17.
In the proof of Corollary 1.52, we observe that hyperconnectedmorphisms are precise, so that any supercompactly generated two-valued topos issupercompact. Taking C to be any non-trivial poset with a maximal element andtaking E to be the category of presheaves on this poset provides an example of asupercompactly generated, supercompact topos which is not two-valued. Example 3.18.
The objects of a reductive category need not be supercompactwithin this category, in spite of Corollary 1.15. For example, consider the four-element lattice: 1 a b a ← → b is 1, which is to say that the arrows from a and b to 1 form a strictly epicfamily containing no strict epimorphism. Even if we relax to mere epimorphismshere, the empty family is strictly epic over 0 yet has no inhabited subfamilies.By considering the lattice of subsets of N as a distributive join semilattice, wesimilarly find that objects of coalescent categories need not be compact in thosecategories. xample 3.19. A familiar example of a localic, locally connected topos which isnot supercompactly (or even compactly) generated is the topos of sheaves on thereal numbers: no non-trivial open sets in the reals are compact. Sh( R ) is not totallyconnected, so by Lemma 2.50 none of its points are relatively precise. Example 3.20.
As a more original non-example, we construct a localic, totallyconnected topos which is not compactly generated. Consider the poset P whose ob-jects are the natural numbers (excluding 0), and with order given by n ≤ m iff n isdivisible by m , so that 1 is terminal. Endow this poset with the Grothendieck topol-ogy J whose covering sieves on a natural number n are those containing cofinitelymany prime multiples of n .All of these sieves are connected and effective epimorphic; that is, ( P, J ) is alocalic, locally connected, subcanonical site. For every n , ℓ ( n ) is therefore an in-decomposable subterminal object of Sh( P, J ). Since P has finite limits, the toposSh( P, J ) is moreover totally connected. To show that Sh(
P, J ) fails to be compactlygenerated it suffices to show that none of the ℓ ( n ) are. But by construction each ℓ ( n ) has a nontrivial infinite covering family by other representables which containsno finite subcovers. Thus this topos has no supercompact objects, and the onlycompact object is the initial object. Example 3.21.
As promised earlier, we demonstrate that it is not possible toextend Theorem 1.53(i) or (ii) to relatively precise or relatively proper surjections.Consider the poset P constructed as a fractal tree with countably many roots andbranches. Explicitly, it has elements non-empty finite sequences of natural numbers, ` ∞ n =1 N n , with ~x ≤ ~y if ~y is an initial segment of the ~x . The Alexandroff locale L corresponding to P has opens which are downward-closed subsets in this ordering,so that for any sequence ~x in an open set, all extensions of ~x also lie in that open.Consider the collection of opens U such that if ( x , . . . , x k − , x k ) ∈ U , then( x , . . . , x k − , y ) ∈ U for cofinitely many values of y . This collection is clearly closedunder finite intersections and arbitrary unions (we needed the sequences in P to benon-empty to ensure that the empty intersection of opens was included here), whichmakes it a subframe of O ( L ). This corresponds to some locale L ′ such that there is asurjective locale map L → L ′ , and hence a geometric surjection s : Sh( L ) → Sh( L ′ ).Moreover, this surjection is relatively precise. Indeed, if X is a sheaf on L ′ and weare given a covering of s ∗ ( X ) in Sh( L ), we may without loss of generality assumethat s ∗ ( X ) is covered by supercompact opens of L , and each supercompact opencontains the open of L ′ consisting of the strict extensions of sequences it contains; X is necessarily the union of these in Sh( L ′ ).However, Sh( L ′ ) is not supercompactly or even compactly generated, since theopens of the form s ∗ ( U ) are not compact, with the exception of the initial open,despite s ∗ preserving any supercompact objects which exist. Bibliography [1] J. B´enabou. Introduction to Bicategories.
Reports of the Midwest CategorySeminar. Lecture Notes in Mathematics , 47, 1967.[2] P. Bridge.
Essentially Algebraic Theories and Localizations in Toposes andAbelian Categories . PhD thesis, University of Manchester, 2012.
3] O. Caramello. A Topos-theoretic Approach to Stone-type Dualities. arXiv:math.CT/1006.3930 , 2011.[4] O. Caramello. Site Characterizations for Geometric Invariants of Toposes.
The-ory and Applications of Categories , 26, 2012.[5] O. Caramello. Syntactic Characterizations of Properties of Classifying Toposes.
Theory and Applications of Categories , 26, 2012.[6] O. Caramello. Topological Galois Theory.
Advances in Mathematics , 291, 2016.[7] O. Caramello.
Theories, Sites, Toposes: Relating and studying mathematicaltheories through topos-theoretic ‘bridges’ . Oxford University Press, 2017.[8] O. Caramello. Denseness Conditions, Morphisms and Equivalences of Toposes. arXiv:math.CT/1906.08737 , 2019.[9] S. A. Celani and I. Calomino. Some Remarks on Distributive Semilattices.
Commentationes Mathematicae Universitatis Carolinae , 54, 2013.[10] Celani, S. A. Topological Representation of Distributive Semilattices.
ScientiaeMathematicae Japonicae Online , 8, 2002.[11] P. Deligne.
SGA4, ´Expos´ee VI, appendix . Seminaire de Geometrie Algebriquedu Bois-Marie, 1963.[12] S. I. Gelfand and Yu. I. Manin.
Methods of Homological Algebra, second edition .Springer-Verlag Berlin Heidelberg, 2003.[13] G. Gr¨atzer.
General Lattice Theory, second edition . Springer Science andBusiness Media, 2002.[14] P. T. Johnstone.
Topos Theory . Academic Press Inc. (London) Ltd., 1977.[15] P. T. Johnstone.
Stone Spaces . Cambridge University Press, 1982.[16] P. T. Johnstone.
Sketches of an Elephant: A Topos Theory Compendium,volumes 1 and 2 . Clarendon Press Oxford, 2002.[17] S. Kondo and S. Yasuda. Sites whose Topoi are the Smooth Representationsof Locally Prodiscrete Monoids. arXiv:1506.08023 , 2017.[18] S. Mac Lane and I. Moerdijk.
Sheaves in Geometry and Logic . Springer-Verlag,1992.[19] I. Moerdijk and J.C.C. Vermeulen. Relative Compactness Conditions forToposes.
Utrecht University Repository (preprint) , 1997.[20] M. Rogers. Toposes of Topological Monoid Actions.
TBC , 2021., 2021.