aa r X i v : . [ m a t h . C T ] O c t FORMAL MODEL THEORY & HIGHER TOPOLOGY
IVAN DI LIBERTI † Abstract.
We study the 2-categories of (generalized) bounded ionads BIonand accessible categories with directed colimits Acc ω as an abstract frameworkto approach formal model theory. We relate them to topoi and (lex) geometricsketches, which serve as categorical specifications of geometric theories. Weprovide reconstruction and completeness-like results. We relate abstract ele-mentary classes to locally decidable topoi, and categories of saturated objectsto atomic topoi. Contents
Introduction 11. Generalized axiomatizations and the Scott construction 42. Classifying topoi 63. Abstract elementary classes and locally decidable topoi 114. Categories of saturated objects, atomicity and categoricity 14Appendix A. Accessible and locally presentable categories 17Appendix B. Sketches 19Appendix C. Topoi 20Appendix D. Ionads 22Acknowledgements 27References 27
Introduction
In the late 1980’s Makkai and Par´e presented their book
Accessible categories: thefoundations of categorical model theory [MP89], providing a solid framework thatcould accommodate a large portion of categorical logic. In the fashion of abstractlogic and abstract model theory, the book has two main aspects: one semanticaland one syntactic. On the one hand they introduced the theory of accessible cate-gories , these are abstract categories of models of some theory. On the other handthey present the theory of sketches , which provide a categorical specification ofinfinitary first order theories. The interplay between sketches (syntax) and acces-sible categories (semantics) is a large portion of categorical model theory. Sincethen, categorical model theory has evolved significantly, thanks to the contribu-tion of several authors, including the authors of the above mentioned book. Thestudy of accessible categories from the point of view of the model theorist has ledto the individuation of special classes of accessible categories, that best suit the † This research was mostly developed during the PhD studies of the author and has beensupported through the grant 19-00902S from the Grant Agency of the Czech Republic. Thefinalization of this research has been supported by the GACR project EXPRO 20-31529X andRVO: 67985840. Which had already appeared under a different name in the work of Lair and Rosick´y Which had been developed by the French school. † most natural constructions of model theory. Among the most common additionalrequirements, we find: • the existence of directed colimits; • amalgamation property (AP); • joint embedding property (JEP); • every morphism is a monomorphism; • the existence of a (very) well behaved faithful functor A → Set preservingdirected colimits.Each of these different assumptions is motivated by some model theoretic intuition.For example, the request that every morphism is a monomorphism is motivated bythe focus on elementary embedding, rather than homomorphisms of structures. Thefaithful functor into
Set allows to construct directed colimits of models as colimitsof underlying structures. The combination of (AP), (JEP) and the existence ofdirected colimits allows the construction of saturated objects [Ros97]. Synthetizingthe conjoint work of Beke, Rosick´y, Lieberman, Vasey et al. (see for example [BR12,LRV17, LR15, LRV17, LR14, LRV18, Vas19a]) in a sentence, accessible categorieswith directed colimits generalize Shelah’s framework of abstract elementary classes,and are special enough to recover the main features of categorical model theory.This paper is concerned with formal model theory , in two specific incarnations. Onthe one hand we intend to study the 2-category of accessible categories withdirected colimits , where 1-cells are functors preserving directed colimits and 2-cells are natural transformations, Acc ω . This is coherent with the classical tradition `a la Makkai-Par´e, and the additionalassumptions that we have listed above, will re-emerge in this setting, dependingon the kind of constructions and behavior typical of model theory that we wantto simulate. On the other hand we introduce a model theoretic perspective on(generalized) bounded ionads, BIon . The first notion of ionads was introduced by Garner [Gar12], mainly from a topolog-ical point of view. In this paper we introduce the notion of ionad of models of ageometric theory, and we give a ionadic interpretation of Makkai’s Ultracategories.On the syntactic side of this paper we find topoi and (lex) geometric sketches ,these are both categorical specifications of geometric theories, as we discuss in thededicated appendices.
LGSketches
Acc ω BIon Topoi ג Mod M od ST S O pt pt ORMAL MODEL THEORY & HIGHER TOPOLOGY 3
From a technical point of view we build on two previous paper of ours [Lib20a,Lib20b], where we develop relationships between topoi, bounded ionads, and acces-sible categories with directed colimits.LocTop Pos ω pt pt O SST
TopoiBIon Acc ω pt pt O SST [Lib20b] shows that these results categorify the Scott topology on a poset withdirected joins and the Isbell duality between locales and topological spaces. Wewill briefly recall the results on those papers in the first section, contextualizingthem in the framework of Lawvere functorial semantics. In interaction betweenhigher topology and completeness-like theorems places this paper in the realm ofStone dualities.
Achtung!.
The paper has four appendices, the first three of them are completelyexpository. We believe that a challenging tasks in reading this paper is the scientificbackground. The model theorist might face for the first time sketches, accessiblecategories, and topoi, while topos theorists might not be completely acquaintedwith accessible categories (and vice-versa). For this reason, we dedicate a shortappendix to each of these three topics, providing references and comments. Thelast appendix is not expository, it introduces the reader to ioands, and contains acouple of technical novelties that are relevant in the discussion.
Structure.
The exposition is organized as follows:Sec. 1 The first section recalls the most relevant results of [Lib20b], the
Scottadjunction and the categorified Isbell duality , putting them in thecontext of functorial semantics. We traces back the Scott topos to theseminal works of Linton and Lawvere on algebraic theories and algebraicvarieties.Sec. 2 The second section inspects a very natural guess that might pop up in themind of the topos theorist: is there any relation between Scott topoi and classifying topoi ? The question will have a partially affirmative answer inthe first subsection. The second one subsumes these partial results. Indeedevery theory S has a category of models Mod ( S ), but this category doesnot retain enough information to recover the theory, even when the theoryhas enough points. That’s why the Scott adjunction is not sharp enough.Nevertheless, every theory has a ionad of models M od ( S ), the category ofopens of such a ionad OM od ( S ) recovers theories with enough points.Sec. 3 This section describes the relation between the Scott adjunction and ab-stract elementary classes, providing a restriction of the Scott adjunction toone between accessible categories where every map is a monomorphism andlocally decidable topoi.Sec. 4 In this section we give the definition of category of saturated objects (CSO)and show that the Scott adjunction restricts to an adjunction betweenCSO and atomic topoi. This section can be understood as an attempt toconceptualize the main result in [Hen19]. Notations and conventions.
Most of the notation will be introduced whenneeded and we will try to make it as natural and intuitive as possible, but wewould like to settle some notation.
IVAN DI LIBERTI † (1) A , B will always be accessible categories, very possibly with directed col-imits.(2) X , Y will always be ionads.(3) Ind λ is the free completion under λ -directed colimits.(4) A κ is the full subcategory of κ -presentable objects of A .(5) G , T , F , E will be Grothendieck topoi.(6) In general, C is used to indicate small categories.(7) η is the unit of the Scott adjunction.(8) ǫ is the counit of the Scott adjunction.(9) ¶ ( X ) is the category of small copresheaves of X .(10) An Isbell topos is a topos of the form O ( X ), for some bounded ionad X ;(11) A Scott topos is a topos of the form S ( A ) for some accessible category A with directed colimits. Notation 0.0.1 (Presentation of a topos) . A presentation of a topos G is the dataof a geometric embedding into a presheaf topos f ∗ : Set C ⇆ G : f ∗ . This meansprecisely that there is a suitable topology τ f on C that turns G into the categoryof sheaves over τ ; in this sense f presents the topos as the category of sheaves overthe site ( C, τ f ).1. Generalized axiomatizations and the Scott construction
Remark 1.0.1.
Let
Grp be the category of groups and U : Grp → Set be theforgetful functor. The historical starting point of a categorical understanding ofuniversal algebra was precisely that one can recover the (a maximal presentationof) the algebraic theory of groups from U . Consider all the natural transformationsof the form µ : U n ⇒ U m , these can be seen as implicitly defined operations of groups. If we gather theseoperations in an equational theory T U , we see that the functor U lifts to the categoryof models Mod ( T U ) as indicated by the diagram below. Grp Mod ( T U ) Set U |−| It is a quite classical result that the comparison functor above is fully faithful andessentially surjective, thus we have axiomatized the category of groups (probablywith a non minimal family of operations).
Remark 1.0.2.
The idea above was introduced in Lawvere’s PhD thesis [Law63]and later developed in great generality by Linton [Lin66, Lin69]. The interestedreader might find interesting [AR94][Chap. 3] and the expository paper [HP07].Nowadays this is a standard technique in categorical logic and some generalizationsof it were presented in [Ros81] by Rosick´y and later again in [LR14][Rem. 3.5].
Remark 1.0.3 (Lieberman-Rosick´y construction) . In [LR14][Rem. 3.5] given acouple ( A , U ) where A is an a accessible category with directed colimits togetherwith a faithful functor U : A → Set preserving directed colimits, the authors form acategory U whose objects are finitely accessible sub-functors of U n and whose arrowsare natural transformations between them. Of course there is a naturally attachedsignature Σ U and a naturally attached first order theory T U . In the same fashion asthe previous remarks one finds a comparison functor A → Σ U -Str. In [LR14][Rem.3.5] the authors stress that is the most natural candidate to axiomatize A . A model ORMAL MODEL THEORY & HIGHER TOPOLOGY 5 of T U is the same as a functor U → Set preserving products and subobjects. Ofcourse the functor A → Σ U -Str factors through Mod( U ) (seen as a sketch) l : A → Mod( U ) , but in [LR14][Rem. 3.5] this was not the main concern of the authors. Remark 1.0.4 (Rosick´y’s remark) . Rem. 1.0.1 ascertains that the collection offunctor { U n } n ∈ N , together with all the natural transformations between them, re-tains all the informations about the category of groups. Observe that in this specificcase, the functors U n all preserve directed colimits, because finite limits commutewith directed colimits. More generally, when A does not come equipped with aspecial forgetful functor, or simply we don’t want to choose a specific one, wecould follow the general strategy of the remarks above and collect all the functorspreserving directed colimits into Set in a category. This is the Scott construcion.
Construction 1.0.5 (The Scott construction) . We recall the construction of S from [Hen19] and [Lib20a]. Let A be an accessible category with directed colimits. S ( A ) is defined as the category the category of functors preserving directed colimitsinto sets. S ( A ) = Acc ω ( A , Set ) . The category S ( A ) is a Grothendieck topos and thus can be seen as a geometrictheory. Following the discussion above, this is a candidate geometric axiomatizationof A . In [Lib20a] we study the Scott construction and show that it is functorial,providing a left adjoint for the functor of points. Remark 1.0.6 (The functor pt ) . The functor of points pt : Topoi → Acc ω be-longs to the literature since quite some time, pt is the covariant hom functorTopoi( Set , − ). It maps a Grothendieck topos G to its category of points, G Cocontlex( G , Set ) . Of course given a geometric morphism f : G → E , we get an induced morphism pt ( f ) : pt ( G ) → pt ( E ) mapping p ∗ p ∗ ◦ f ∗ . The fact that Topoi( Set , G ) isan accessible category with directed colimits appears in the classical reference byBorceux as [Bor94a][Cor. 4.3.2], while the fact that pt ( f ) preserves directed colimitsfollows trivially from the definition. Remark 1.0.7.
When we idenitify the category of topoi with a localization of thecategory of geometric theories, the functor of points is computing the (set theoretic)models of the theory classified by the topos. Being a right adjoint, it is coherentwith the intuition that its left adjoint computes the free theory over an accessiblecategory with directed colimits.
Theorem 1.0.8 ([Hen19][Prop. 2.3],[Lib20a][Thm. 2.1] The Scott adjunction) . The 2-functor of points pt : Topoi → Acc ω has a left biadjoint S , yielding the Scottbiadjunction, S : Acc ω ⇆ Topoi : pt . Remark 1.0.9 (Rosick´y’s remark) . Going back to Rosick´y-Lieberman construc-tion, the previous discussion implies that the small category { U n } n ∈ N is a fullsubcategory of the Scott topos of the category of groups. In fact the vocabulary ofthe theory that we used to axiomatize the category of groups is made up of symbolscoming from a full subcategory of the Scott topos. Remark 1.0.10 (Generalized axiomatizations) . The generalized axiomatization ofLieberman and Rosick´y amounts to a sketch U . As we mentioned, there exists anobvious inclusion of U in the Scott topos of A , i : U → S ( A ) IVAN DI LIBERTI † which is a flat functor because finite limits in S ( A ) are computed pointwise in Set A .Thus, every point p : Set → S ( A ) induces a model of the sketch U by composition, i ∗ : pt ( S A ) → Mod( U ) p p ∗ ◦ i. In particular this shows that the unit of the Scott adjunction lifts the comparisonfunctor between A and Mod( U ) along i ∗ and thus the Scott topos provides a sharper axiomatization of T U . A ptS ( A ) Mod( U ) η A li ∗ Remark 1.0.11 (Faithful functors are likely to generate the Scott topos) . Yet, itshould be noticed that when U is a generator in S ( A ), the functor i ∗ is an equivalenceof categories. As unlikely as it may sound, in all the examples that we can thinkof, a generator of the Scott topos is always given by a faithful forgetful functor U : A → Set . This phenomenon is so pervasive that the author has believed forquite some time that an object in the Scott topos S ( A ) is a generator if and only ifit is faithful and conservative. We still lack a counterexample, or a theorem provingsuch a statement. 2. Classifying topoi
This section is devoted to specifying the connection between Scott topoi, Isbelltopoi and classifying topoi. Recall that for a geometric theory T , a classifyingtopos Set [ T ] is a topos representing the functor of models in topoi, Mod ( − ) ( T ) ∼ = Topoi( − , Set [ T ]) . The theory of classifying topoi allows us to internalize geometric logic in the internallogic of the 2-category of topoi. The reader that is not familiar with the theory ofclassifying topoi is encouraged to check the Appendix.2.1.
Categories of models, Scott topoi and classifying topoi.
The Scotttopos S ( Grp ) of the category of groups is
Set
Grp ω , this follow from [Lib20a][Rem2.13] and applies to Mod ( T ) for every Lawvere theory T . It is well known that Set
Grp ω is also the classifying topos of the theory of groups. This section is devotedto understating if this is just a coincidence, or if the Scott topos is actually relatedto the classifying topos. Remark 2.1.1.
Let A be an accessible category with directed colimits. In orderto properly ask the question is S ( A ) the classifying topos? , we should answer thequestion the classifying topos of what? Indeed A is just a category, while one cancompute classifying topoi of theories. Our strategy is to introduce a quite generalnotion of theory that fits in the following diagram,Acc ω Topoi
Theories
SptMod ( − ) ג ( − ) in such a way that: ORMAL MODEL THEORY & HIGHER TOPOLOGY 7 (1) ג ( T ) gives the classifying topos of T ;(2) Mod ( − ) ∼ = pt ג ( − ).In this new setting we can reformulate our previous discussion in the followingmathematical question: ג ( − ) ? ∼ = SMod ( − ) . Remark 2.1.2 (Geometric Sketches) . The notion of theory that we plan to useis that of geometric sketch. The category of (small) sketches was described in[MP89][3.1], while a detailed study of geometric sketches was conducted in [AJMR97,AR96]. Acc ω Topoi
GSketches
SptMod ( − ) ג ( − ) Remark 2.1.3.
Following [MP89], there exists a natural way to generate a sketchfrom any accessible category. This construction, in principle, gives even a leftadjoint for the functor
Mod ( − ), but does land in large sketches. Thus it is indeedtrue that for each accessible category there exist a sketch (a theory) canonicallyassociated to it. We do not follow this line because the notion of large sketch,from a philosophical perspective, is a bit unnatural. Syntax should always be veryfrugal. From an operational perspective, presentations should always be as smallas possible. It is possible to cut down the size of the sketch, but this constructioncannot be defined functorially on the whole category of accessible categories withdirected colimits. Since elegance and naturality is one of the main motivationsfor this treatment of syntax-semantics dualities, we decided to avoid any kind ofnon-natural construction. Remark 2.1.4.
Geometric sketches contain coherent sketches. In the dictio-nary between logic and geometry that is well motivated in the indicated papers([AJMR97, AR96]) these two classes correspond respectively to geometric and co-herent theories. The latter essentially contain all first order theories via the processof Morleyzation. These observations make our choice of geometric sketches a verygeneral notion of theory and makes us confident that it’s a good notion to look at.We now proceed to describe the two functors labeled with the name of
Mod and ג . Remark 2.1.5 (Mod) . This 2-functor is very easy to describe. To each sketch S we associate its category of Set-models, while it is quite evident that a morphismof sketches induces by composition a functor preserving directed colimits (see Sec.B in the Background chapter). Construction 2.1.6 ( ג ) . The topos completion of a geometric sketch is a highlynontrivial object to describe. Among the possible constructions that appear in theliterature, we refer to [Bor94a][4.3]. Briefly, the idea behind this construction is thefollowing.(1) By [Bor94a][4.3.3], every sketch S can be completed to a sketch ¯ S whoseunderlying category is cartesian.(2) By [Bor94a][4.3.6], this construction is functorial and does not change themodel of the sketch in any Grothendieck topos.(3) By [Bor94a][4.3.8], the completion of the sketch has a natural topology ¯ J . IVAN DI LIBERTI † (4) The correspondence S ¯ S ( ¯ S, ¯ J ) transforms geometric sketches intosites and morphism of sketches into morphism of sites.(5) We compute sheaves over the site ( ¯ S, ¯ J ).(6) Define ג to be S ¯ S ( ¯ S, ¯ J ) Sh ( ¯ S, ¯ J ). Remark 2.1.7.
While [Bor94a][4.3.6] proves that
Mod ( − ) ≃ pt ג ( − ), and [Bor94a][4.3.8]prove that ג ( S ) is the classifying topos of S among Grothendieck topoi, the mainquestion of this section remains completely open, is ג ( S ) isomorphic to the Scotttopos SMod ( − ) of the category of Set models of S ? We answer this question withthe following theorem. Theorem 2.1.8.
If the counit ǫ ג ( S ) of the Scott adjunction is an equivalence ofcategories on ג ( S ), then ג ( S ) coincides with SMod ( S ). Proof.
We introduced enough technology to make this proof incredibly slick. Recallthe counit
Spt ( ג ( S )) → ג ( S )and assume that it is an equivalence of categories. Now, since Mod ( − ) ≃ pt ג ( − ),we obtain that ג ( S ) ≃ SMod ( S ) , which indeed it our thesis. (cid:3) Remark 2.1.9. [Lib20b][Thm 5.0.2 and 5.2.3] characterize those topoi for whichthe counit is an equivalence of categories, providing a full description of thosegeometric sketches for which ג ( S ) coincides with SMod ( S ). Since Thm. 2.1.8 mightnot look satisfactory, in the following comment we use [Lib20b][Cor. 5.2.5] to derivea nice looking statement. Corollary 2.1.10.
Assume ג ( S ) has enough points and Mod ( S ) is complete. Let i : ג ( S ) → Set C be a presentation such that pt ( i ) preserve limits. then ג ( S )coincides with SMod ( S ). Proof.
Apply Cor. [Lib20b][Cor. 5.2.5] to Thm. 2.1.8. (cid:3)
Ionads of models, Isbell topoi and classifying topoi.
Indeed the mainresult of this section up to this point has been partially unsatisfactory. As happenssometimes, the answer is not as nice as expected because the question in the firstplace did not take in consideration some relevant factors. The category of modelsof a sketch does not retain enough information on the sketch. Fortunately, we willshow that every sketch has a ionad of models (not just a category) and the categoryof opens of this ionad is a much better approximation of the classifying topos. Inthis subsection, we switch diagram of study to the one below.BIon Topoi
LGSketches O pt M od ( − ) ג ( − ) Of course, in order to study it, we need to introduce all its nodes and legs. Weshould say what we mean by
LGSketches and M od ( − ). The adjunction O ⊣ pt was introduced and studied in [Lib20b] and it relates topoi to bounded ionads, werefer to [Lib20b][Sec. 3] for the construction, while an introduction to ionads canbe found in the Appendix. ORMAL MODEL THEORY & HIGHER TOPOLOGY 9
Whatever
LGSketches and M od ( − ) will be, the main point of the section is to showthat this diagram fixes the one of the previous section, in the sense that we willobtain the following result. Theorem.
The following are equivalent: • ג ( S ) has enough points; • ג ( S ) coincides with OM od ( S ).We decided to present this theorem separately from the previous one because indeeda ionad of models is a much more complex object to study than a category of models,thus the results of the previous section are indeed very interesting, because easierto handle. Example 2.2.1 (Motivating ionads of models: Ultracategories) . We are not com-pletely used to thinking about ionads of models. Indeed a (bounded) ionad is quitecomplex data, and we do not completely have a logical intuition on its interioroperator.
In which sense does the interior operator equip a category of models witha topology?
One very interesting example, that hasn’t appeared in the literatureto our knowledge is the case of ultracategories. Ultracategories where introducedby Makkai in [AF13] and later simplified by Lurie in [Lur]. These objects are thedata of a category A together with an ultrastructure, that is a family of functors Z X : β ( X ) × A X → A . We redirect to [Lur] for the precise definition. In a nutshell, each of these functors R X defines a way to compute the ultraproduct of an X -indexed family of objectsalong some ultrafilter. Of course there is a notion of morphism of ultracategories,namely a functor A → B which is compatible with the ultrastructure [Lur][Def.1.41]. Since the category of sets has a natural ultrastructure, for every ultracategory A one can define Ult( A , Set ) which obviously sits inside
Set A . Lurie observes thatthe inclusion ι : Ult( A , Set ) → Set A preserves all colimits [Lur][War. 1.4.4], and in fact also finite limits (the proof is thesame). In particular, when A is accessible and every ultrafunctor is accessible, theinclusion ι : Ult( A , Set ) → Set A factors through ¶ ( A ) and thus the ultrastructureover A defines a idempotent lex comonad over ¶ ( A ) by the adjoint functor theorem.This shows that every (good enough) accessible ultracategory yields a ionad, whichis also compact in the sense that its category of opens is a compact (coherent)topos. This example is really a step towards a categorified Stone duality involvingcompact ionads and boolean topoi.2.2.1. LGSketches and M od ( − ) . Definition 2.2.2.
A geometric sketch S is lex if its underlying category has finitelimits and every limiting cone is in the limit class. Remark 2.2.3 (Lex sketches are enough ) . [Bor94a][4.3.3] shows that every geo-metric sketch can be replaced with a lex geometric sketch in such a way that theunderlying category of models, and even the classifying topos, does not change. Inthis sense this full subcategory of geometric sketches is as expressive as the wholecategory of geometric sketches. Proposition 2.2.4 ( M od ( − ) on objects) . Every lex geometric sketch S inducesa ionad M od ( S ) over its category of models Mod ( S ). † Proof.
The underlying category of the ionad M od ( S ) is Mod ( S ). We must providean interior operator (a lex comonad),Int S : ¶ ( Mod ( S )) → ¶ ( Mod ( S )) . In order to do so, we consider the evaluation pairing eval : S × Mod ( S ) → Set map-ping ( s, p ) p ( s ). Let ev : S → Set
Mod ( S ) be its mate. Similarly to [Lib20b][Con.3.2.8], such functor takes values in ¶ ( Mod ( S )). Because S is a lex sketch, thisfunctor must preserve finite limits. Indeed, ev (lim s i )( − ) ∼ = ( − )(lim s i ) ∼ = lim(( − )( s i )) ∼ = lim ev ( s i )( − ) . Now, the left Kan extension lan y ev (see diagram below) is left exact because ¶ ( Mod ( S )) is an infinitary pretopos and ev preserves finite limits. S ¶ ( Mod ( S )) Set S ◦ ev y lan y ev Moreover it is cocontinuous because of the universal property of the presheaf con-struction. Because
Set S ◦ is a total category, lan y ev must have a right adjoint (andit must coincide with lan ev y ). The induced comonad must be left exact, becausethe left adjoint is left exact. DefineInt S := lan y ev ◦ lan ev y. Observe that Int S coincides with the density comonad of ev by [Lib19][A.7]. Suchresult dates back to [AT69]. (cid:3) Remark 2.2.5 ( M od ( − ) on morphism of sketches) . This definition will not begiven explicitly: in fact we will use the following remark to show that the ionadabove is isomorphic to the one induced by ג ( S ), and thus there exists a natural wayto define M od ( − ) on morphisms.2.2.2. Ionads of models and theories with enough points.
Remark 2.2.6.
In the main result of the previous section, a relevant rˆole wasplayed by the fact that pt ג ≃ Mod . The same must be true in this one. Thuswe should show that pt ג ≃ M od . Indeed we only need to show that the interioroperator is the same, because the underlying category is the same by the discussionin the previous section.
Proposition 2.2.7. pt ◦ ג ≃ M od . Proof.
Let S be a lex geometric sketch. Of course there is a map j : S → ג S , because S is a site of definition of ג S . Moreover, j is obviously dense. In particular theevaluation functor that defines the ionad pt ◦ ג given by ev ∗ : ג ( S ) → ¶ ( pt ◦ ג ( S ))is uniquely determined by its composition with j . This means that the comonad ev ∗ ev ∗ is isomorphic to the density comonad of the composition ev ∗ ◦ j . Indeed, ev ∗ ev ∗ ∼ = lan ev ∗ ev ∗ ∼ = lan ev ∗ ( lan j ( ev ∗ j )) ∼ = lan ev ∗ j ( ev ∗ j ) . Yet, ev ∗ j is evidently ev , and thus ev ∗ ev ∗ ∼ = Int S as desired. (cid:3) Theorem 2.2.8.
The following are equivalent:
ORMAL MODEL THEORY & HIGHER TOPOLOGY 11 • ג ( S ) has enough points; • ג ( S ) coincides with OM od ( S ). Proof.
By [Lib20b][Thm. 4.0.3], ג ( S ) has enough points if and only if the counit ofthe categorified Isbell duality ρ : O pt ( ג )( S ) → S is an equivalence of topoi. Now,since pt ◦ ג ∼ = M od , we obtain the thesis. (cid:3) Abstract elementary classes and locally decidable topoi
A general discussion.
This section is dedicated to the interaction betweenAbstract elementary classes and the Scott adjunction. Abstract elementary classeswere introduced in the 70’s by Shelah as a framework to encompass infinitary logicswithin the language of model theorist. In principle, an abstract elementary class A should look like the category of models of a first order infinitary theory whosemorphisms are elementary embeddings. The problem of relating abstract elemen-tary classes and accessible categories has been tackled by Lieberman [Lie11], andBeke and Rosick´y [BR12], and lately has attracted the interest of model theoristssuch as Vasey, Boney and Grossberg [BGL + Theorem 3.1.1 ([BR12](5.7)) . A category A is equivalent to an abstract elemen-tary class if and only if it is an accessible category with directed colimits, whosemorphisms are monomorphisms and which admits a full with respect to isomor-phisms and nearly full embedding U into a finitely accessible category preservingdirected colimits and monomorphisms. Definition 3.1.2.
A functor U : A → B is nearly full if, given a commutativediagram, U ( a ) U ( c ) U ( b ) U ( f ) h U ( g ) in B , there is a map ¯ h in A such that h = U (¯ h ) and g ¯ h = f . Observe that when U is faithful such a filling has to be unique. Remark 3.1.3.
In some reference the notion of nearly-full functor was called co-herent, referring directly to the coherence axiom of AECs that it incarnates. Theword coherent is overloaded in category theory, and thus we do not adopt thisterminology, but nowadays it is getting more and more common.
Example 3.1.4 ( pt ( E ) is likely to be an AEC) . Let E be a Grothendieck toposand f ∗ : Set C ⇆ E : f ∗ a presentation of E . By a combination of [Lib20a][Prop.4.2] and [Lib20a][Rem. 2.12], applying the functor pt we get a fully faithful functor pt ( E ) → pt ( Set C ) ∼ = Ind ( C )into a finitely accessibly category. Thus when every map in pt ( E ) is a monomor-phism we obtain that pt ( E ) is an AEC via Thm. 3.1.1. We will see in the nextsection (Thm. 3.2.1) that this happens when E is locally decidable; thus the cate-gory of points of a locally decidable topos is always an AEC. Example 3.1.5 ( η A behaves nicely on AECs) . When A is an abstract elementaryclass, the unit of the Scott adjunction η A : A → ptS ( A ) is faithful and iso-full. Thisfollows directly from [Lib20a][Prop 4.13]. † Remark 3.1.6.
Even if this is the sharpest (available) categorical characterizationof AECs it is not hard to see how unsatisfactory it is. Among the most evidentproblems, one can see that it is hard to provide a categorical understanding ofnearly full and full with respect to isomorphisms. Of course, an other problem isthat the list of requirements is pretty long and very hard to check: when does sucha U exist? It is very hard to understand when such a pseudo monomorphism exists. That iswhy it is very useful to have a testing lemma for its existence.
Theorem 3.1.7 (Testing lemma) . Let A be an object in Acc ω where every mor-phism is a monomorphism. If η A is a nearly-full pseudo monomorphism, then A isan AEC. Proof.
The proof is relatively easy, choose a presentation f ∗ : Set C ⇆ S ( A ) : f ∗ of S ( A ). Now in A η A → ptS ( A ) → pt ( Set C ) ∼ = Ind ( C ) , by a combination of [Lib20a][Prop. 4.2] and [Lib20a][Rem. 2.12], the compositionis a faithful and nearly full functor preserving directed colimits from an accessiblecategory to a finitely accessible category, and thus A is an AEC because of Thm.3.1.1. (cid:3) Locally decidable topoi and AECs.
The main result of this subsectionrelates locally decidable topoi to AECs. The full subcategory of Acc ω whose objectsare AECs will be indicated by AECs. As in the previous chapters, let us give theprecise statement and then discuss it in better detail. Theorem 3.2.1.
The Scott adjunction restricts to locally decidable topoi andAECs. S : AECs ⇆ LDTopoi : pt Locally decidable topoi.
The definition of locally decidable topos will appearobscure at first sight.
Definition 3.2.2 (Decidable object) . An object e in a topos E is decidable if thediagonal map e → e × e is a complemented subobject. Definition 3.2.3 (Locally decidable topos) . An object e in a topos E is calledlocally decidable iff there is an epimorphism e ′ ։ e such that e ′ is a decidableobject. E is locally decidable if every object is locally decidable.In order to make the definition above clear we should really define decidable objectsand discuss their meaning. This is carried out in the literature and it is not ourintention to recall the whole theory of locally decidable topoi. Let us instead givethe following characterization, that we may take as a definition. Theorem 3.2.4 ([Joh02b][C5.4.4], Characterization of loc. dec. topoi) . The fol-lowing are equivalent:(1) E is locally decidable;(2) there exists a site ( C, J ) of presentation where every map is epic;(3) there exists a localic geometric morphism into a Boolean topos.
Remark 3.2.5.
Recall that a localic topos E is a topos of sheaves over a locale.The theorem above (which is due to Freyd [Fre72]) shows that a locally decidabletopos is still a topos of sheaves over a locale, but the locale is not in Set . It isinstead in some boolean topos. A boolean topos is the closest kind of topos we can
ORMAL MODEL THEORY & HIGHER TOPOLOGY 13 think of to the category of sets itself. For more details, we redirect the reader tothe Background chapter, where we give references to the literature.3.2.2.
Proof of Thm. 3.2.1.Proof of Thm. 3.2.1. • Let E be a locally decidable topos. By Exa. 3.1.4, it is enough to showthat every map in pt ( E ) is a monomorphism. This is more or less a folkloreresult, let us give the shortest path to it given our technology. Recall thatone of the possible characterization of a locally decidable topos is that it hasa localic geometric morphism into a boolean topos E → B . If B is a booleantopos, then every map in pt ( G ) is a monomorphism [Joh02b][D1.2.10, lastparagraph]. Now, the induce morphism below, pt ( E ) → pt ( B ) , is faithful by Prop. [Lib20a][Prop. 4.4]. Thus every map in pt E must be amonomorphism. • Let’s show that for an accessible category with directed colimits A , its Scotttopos is locally decidable. By [Joh02b][C5.4.4], it’s enough to prove that S A has a site where every map is an epimorphism. Using [Lib20a][Rem. 2.9], A ◦ κ is a site of definition of S A , and since every map in A is a monomor-phism, every map in A ◦ κ is epic. (cid:3) The previous theorem admits an even sharper version.
Theorem 3.2.6.
Let A be an accessible category with directed colimits and afaithful functor U : A → Set preserving directed colimits. If S A is locally decidable,then every map in A is a monomorphism. Proof.
Step 1 If G is a boolean topos, then every map in pt ( G ) is a monomorphism[Joh02b][D1.2.10, last paragraph].Step 2 Recall that one of the possible characterization of a locally decidable toposis that it has a localic geometric morphism into a boolean topos S ( A ) → G .Step 3 In the following diagram A η A → ptS ( A ) → pt ( G ) , the composition is a faithful functor by [Lib20a][Prop. 4.4 and 4.13]. Thus A has a faithful functor into a category where every map is a monomor-phism. As a result every map in A is a monomorphism. (cid:3) Remark 3.2.7.
The following corollary gives a complete characterization of thosecontinuous categories that are abstract elementary classes. Recall that continuouscategories were defined in [JJ82] in analogy with continuous posets in order to studyexponentiable topoi. Among the possible characterizations, a category is continuousif and only if it is a reflective subcategory of a finitely accessible category whoseright adjoint preserve directed colimits. We discussed continuous categories in thefirst section of [Lib20a].
Corollary 3.2.8 (Continuous categories and AECs) . Let A be a continuous cate-gory. The following are equivalent:(1) A is an AEC.(2) Every map in A is a monomorphism.(3) S ( A ) is locally decidable. † Proof.
Since it’s a split subobject in Acc ω of a finitely accessible category, thehypotheses of [BR12][5.7] are met. (cid:3) Categories of saturated objects, atomicity and categoricity
Remark 4.0.1.
In this section we define categories of saturated objects and studytheir connection with atomic topoi and categoricity. The connection between atomictopoi and categoricity was pointed out in [Car12]. This section corresponds to akind of syntax-free counterpart of [Car12]. In the definition of category of saturatedobjects we axiomatize the relevant properties of the inclusion ι : Set κ → Set andwe prove the following two theorems.
Theorem. (1) If A is a category of saturated objects, then S ( A ) is an atomic topos.(2) If in addition A has the joint embedding property, then S ( A ) is booleanand two valued.(3) If in addition η A is isofull and faithful and surjective on objects, then A iscategorical in some presentability rank. Theorem. If E is an atomic topos, then pt ( E ) is a candidate category of saturatedobjects.Let us recall (or introduce) the notion of ω -saturated object in an accessible categoryand the joint embedding property. Definition 4.0.2.
Let A be an accessible category. We say that s ∈ A is ω -saturated if it is injective with respect to maps between finitely presentable objects.That is, given a morphism between finitely presentable objects f : p → p ′ and amap p → s , there exists a lift as in the diagram below. sp p ′ Remark 4.0.3.
In general, when we look at accessible categories from the per-spective of model theory, every map in A is a monomorphism, and this definitionis implicitly adding the hypothesis that every morphism is injective . Remark 4.0.4.
A very good paper to understand the categorical approach tosaturation is [Ros97].
Definition 4.0.5.
Let A be a category. We say that A has the joint embeddingproperty if given two objects A, B there exist and object C and two morphisms A → C , B → C . Remark 4.0.6.
In [Hen19], Henry proves that there are AECs that cannot appearas the category of points of a topos, which means that they cannot be axiomatizedin L ∞ ,ω . This answers a question initially asked by Rosick´y at the conference Cat-egory Theory 2014 and makes a step towards our understanding of the connectionbetween accessible categories with directed colimits and axiomatizable classes. Themain tool that allows him to achieve this result is called in the paper the Scottconstruction ; he proves the Scott topos of
Set ≥ κ is atomic. Even if we developedtogether the relevant rudiments of the Scott construction, the reason for which thisresult was true appeared to the author of this thesis enigmatic and mysterious.With this motivation in mind we came to the conclusion that the Scott topos The category of sets of cardinality at least κ and injective functions The author of this thesis.
ORMAL MODEL THEORY & HIGHER TOPOLOGY 15 of Set ≥ κ is atomic because of the fact that Set ≥ κ appears as a subcategory ofsaturated objects in Set . Remark 4.0.7.
As a direct corollary of the theorems in this section one gets backthe main result of [Hen19], but this is not the main accomplishment of this section.Our main contribution is to present a conceptual understanding of [Hen19] and aneat technical simplification of his proofs. We also improve our poor knowledge ofthe Scott adjunction, trying to collect and underline its main features. We feel thatthe Scott adjunction might serve as a tool to have a categorical understanding ofShelah’s categoricity conjecture for accessible categories with directed colimits.
Remark 4.0.8 (What is categoricity and what about the categoricity conjecture?) . Recall that a category of models of some theory is categorical in some cardinality κ if it has precisely one model of cardinality κ . Morley has shown in 1965 that if acategory of models is categorical in some cardinal κ , then it must be categorical inany cardinal above and in any cardinal below up to ω ([CK90]). We will be moreprecise about Morley’s result in the section about open problems. When Abstractelementary classes were introduced in the 1970’s, Shelah chose Morley’s theoremas a sanity check result for his definition. Since then, many approximations ofthese results has appeared in the literature. The most updated to our knowledgeis contained in [Vas19b]. We recommend the paper also as an introduction to thistopic. Definition 4.0.9 ((Candidate) categories of ( ω -)saturated objects) . Let A be acategory in Acc ω . We say that A is a category of (finitely) saturated objects ifthere a is topological embedding j : A → K in Acc ω such that:(1) K is a finitely accessible category.(2) j A ⊂ Sat ω ( K ) .(3) K ω has the amalgamation property .We say that A is a candidate category of (finitely) saturated objects if there existsa functor j that verifies (1)-(3). Remark 4.0.10.
The notion of category of saturated objects axiomatizes the prop-erties of the inclusion j : Sat ω ( K ) ֒ → K , our motivating example was the inclusionof Set ≥ κ ֒ → Set ≥ ω ֒ → Set . The fact that every object in
Set ≥ κ is injective withrespect to finite sets is essentially the axiom of choice. [Ros97] describes a directconnection between saturation and amalgamation property, which was also impliedin [Car12].In [Car12], Caramello proves - essentially - that the category of points of an atomictopos is a category of saturated objects and she observes that it is countable cate-gorical. This shows that there is a deep connection between categoricity, saturationand atomic topoi. We recall the last notion before going on with the exposition. Definition 4.0.11 (Characterization of atomic topoi, [Joh02b][C3.5]) . Let G be aGrothendieck topos, then the following are equivalent:(1) G is atomic.(2) G is the category of sheaves over an atomic site.(3) The subobject lattice of every object is a complete atomic boolean algebra.(4) Every object can be written as a disjoint union of atoms. Theorem 4.0.12. (1) If A is a category of saturated objects, then S ( A ) is an atomic topos. The full subcategory of ω -saturated objects. A category has the amalgamation property is every span can be completed to a square. † (2) If in addition A has the joint embedding property, then S ( A ) is booleanand two valued.(3) If in addition η A is iso-full, faithful and surjective on objects, then A iscategorical in some presentability rank. Proof. (1) Let A be a category of saturated objects j : A → K . We must show that S ( A ) is atomic. The idea of the proof is very simple; we will show that:(a) S j presents A as j ∗ : Set K ω ⇆ S ( A ) : j ∗ ;(b) The induced topology on K ω is atomic.(a) follows directly from the definition of topological embedding and [Lib20a][Rem.2.13]. (b) goes identically to [Hen19][Cor. 4.9]: note that for any map k → k ′ ∈ K ω , the induced map j ∗ yk → j ∗ yk ′ is an epimorphism: indeedany map k → ja with a ∈ A can be extended along k → k ′ because j makes A a category of saturated objects. So the induced topology on K ω is the atomic topology (every non-empty sieve is a cover). The fact that K ω has the amalgamation property is needed to make the atomic topologya proper topology.(2) Because A has the joint embedding property, its Scott topos is connected.Indeed a topos is connected when the inverse image of the terminal map t : S ( A ) → Set is fully faithful. t appears as the S ( τ ), where τ is theterminal map τ : A → · . When A has the JEP, and thus is connected, τ is a lax-epi, and f ∗ is fully faithful by [Lib20a][Prop. 4.6]. Then, S ( A ) isatomic and connected. By [Car18][4.2.17] it is boolean two-valued.(3) This follows from [Lib20a][Prop 4.13] and [Car12]. In fact, Caramello hasshown that ptS ( A ) must be countably categorical and the countable objectis saturated (by construction). Thus, the unit of the Scott adjunction mustreflect the (essential) unicity of such an object. (cid:3) Theorem 4.0.13. If E is an atomic topos, then pt ( E ) is a candidate category ofsaturated objects. Proof.
Let E be an atomic topos and i : E → Set C be a presentation of E byan atomic site. It follows from [Car12] that pt ( i ) presents pt ( E ) as a candidatecategory of saturated objects. (cid:3) Categories of κ -saturated objects. Obviously the previous definitions canbe generalized to the κ -case of the Scott adjunction, obtaining analogous results.Let us boldly state them. Definition 4.1.1 ((Candidate) categories of ( κ -)saturated objects) . Let A be acategory in Acc κ . We say that A is a category of κ -saturated objects if there istopological embedding (for the S κ -adjunction) j : A → K in Acc κ such that:(1) K is a κ -accessible category.(2) j A ⊂ Sat κ ( K ).(3) K κ has the amalgamation property.We say that A is a candidate category of κ -saturated objects if there exists a functor j that verifies (1)-(3). Theorem 4.1.2. (1) If A is a category of κ -saturated objects, then S κ ( A ) is an atomic κ -topos.(2) If in addition A has the joint embedding property, then S κ ( A ) is booleanand two valued. ORMAL MODEL THEORY & HIGHER TOPOLOGY 17 (3) If in addition η A is iso-full, faithful and surjective on objects, then A iscategorical in some presentability rank. Theorem 4.1.3. If E is an atomic κ -topos, then pt κ ( E ) is a candidate category of κ -saturated objects. Appendix A. Accessible and locally presentable categories
Achtung! A.0.1.
In this section λ is a regular cardinal. Definition A.0.2 ( λ -accessible category) . A λ -accessible category A is a categorywith λ -directed colimits with a set of λ -presentable objects that generate by λ -directed colimits. An accessible category is a category that is λ -accessible for some λ . Definition A.0.3 (Locally λ -presentable category) . A locally λ -presentable cat-egory is a cocomplete λ -accessible category. A locally presentable category is acategory that is locally λ -presentable for some λ . Definition A.0.4 ( λ -presentable object) . An object a ∈ A is λ -presentable if itscovariant hom-functor A ( a, − ) : A → Set preserves λ -directed colimits. Definition A.0.5 ( λ -directed posets and λ -directed colimits) . A poset P is λ -directed if it is non empty and for every λ -small family of elements { p i } ⊂ P ,there exists an upper bound. A λ -directed colimit is the colimit of a diagram overa λ -directed poset (seen as a category). Notation A.0.6.
For a category A , we will call A λ its full subcategory of λ -presentable objects.A.1. Literature.
There are two main references for the theory of accessible andlocally presentable categories, namely [AR94] and [MP89]. The first one is intendedfor a broader audience and appeared few years after the second one. The second oneis mainly concerned with the logical aspects of this theory. We mainly recommend[AR94] because it appears a bit more fresh in style and definitely less demandingin general knowledge of category theory. A more experienced reader (in categorytheory) that is mainly interested in logic could choose [MP89]. Even though [AR94]treats some 2-categorical aspects of this topic, [MP89]’s exposition is much morecomplete in this direction. Another good general exposition is [Bor94b][Chap. 5].A.2.
A short comment on these definitions.Remark A.2.1.
The theory of accessible and locally presentable categories hasgained quite some popularity along the years because of its natural ubiquity. Mostof the categories of the working mathematician are accessible, with a few (but stillextremely important) exceptions. For example, the category
Top of topologicalspaces is not accessible. In general, categories of algebraic structures are locally ℵ -presentable and many relevant categories of geometric nature are ℵ -accessible.A sound rule of thumb is that locally finitely presentable categories correspond tocategories of models essentially algebraic theories, in fact this is even a theoremin a proper sense [AR94][Chap. 3]. A similar intuition is available for accessiblecategories too, but some technical price must be paid [AR94][Chap. 5]. Accessibleand locally presentable categories (especially the latter) are tame enough to makemany categorical wishes come true; that’s the case for example of the adjoint functortheorem, that has a very easy to check version for locally presentable categories. the unit of the κ -Scott adjunction. This means that its cardinality is strictly less then λ . For example ℵ -small means finite. † Remark A.2.2.
All in all, an accessible category should be seen as a categoryequipped with a small set of small objects such that every object can be obtainedas a kind of directed union of them. In the category of topological spaces, thesesmall objects are not enough to recover any other object from them.
Example A.2.3.
To clarify the previous remark, we give list of locally ℵ -presentablecategories. On the right column we indicate the full subcategory of finitely pre-sentable objects. K K ω Set finite sets
Grp finitely presentable groups
Mod ( R ) finitely presentable modulesIt is not surprising at all that a set X is the directed union of its finite subsets. Remark A.2.4.
Accessible and locally presentable categories have a canonicalrepresentation , in terms of free completions under λ -directed of colimits . This theoryis studied in [AR94][Chap. 2.C]. The free completion of a category C under λ -directed colimits is always indicated by Ind λ ( C ) in this paper. Theorem A.2.5. A λ -accessible category A is equivalent to the free completionof A λ under λ -directed colimits, A ≃ Ind λ ( A λ ) . Remark A.2.6.
Explicit descriptions of the free completion of a category under λ -directed colimits are indeed available. To be more precise, given a category C onecan describe Ind ( C ) as the category of flat functors Flat( C ◦ , Set ). In the specialcase of a category with finite colimits, we have a simpler description of flat functors.Let us state the theorem in this simpler case for the sake of simplicity.
Theorem A.2.7.
Let C be a small category with finite colimits. Its free completionunder directed colimits is given by the category of functors preserving finite limitsfrom C ◦ into sets: Ind ( C ) ≃ Lex ( C ◦ , Set ) . A.3.
Locally presentable categories and essentially algebraic theories.
The connection between locally presentable categories and essentially algebraictheories is made precise in [AR94][Chap. 3]. While algebraic theories axiomatizeoperational theories, essentially algebraic theories axiomatize operational theorieswhose operations are only partially defined. Category theorists have an equivalentapproach to essentially algebraic theories via categories with finite limits. This ap-proach was initially due to Freyd [Fre02], though a seminal work of Coste [Cos76]should be mentioned too.A.4.
Accessible categories and (infinitary) logic.
Accessible categories havebeen connected to (infinitary) logic in several (partially independent) ways. Thisstory is recounted in Chapter 5 of [AR94]. Let us recall two of the most importantresults of that chapter.(1) As locally presentable categories, accessible categories are categories ofmodels of theories, namely basic theories [AR94][Def. 5.31, Thm. 5.35].(2) Given a theory T in L λ the category Elem λ ( T ) of models and λ -elementaryembeddings is accessible [AR94][Thm. 5.42]. ORMAL MODEL THEORY & HIGHER TOPOLOGY 19
Unfortunately, it is not true in general that the whole category of models andhomomorphisms of a theory in L λ is accessible. It was later shown by Lieberman[Lie09] and independently by Rosick´y and Beke [BR12] that abstract elementaryclasses are accessible too. The reader that is interested in this connection mightfind interesting [Vas19a], whose language is probably the closest to that of a modeltheorist. Appendix B. Sketches
Definition B.0.1 (Sketch) . A sketch is a quadruple S = ( S, L, C, σ ) where S is a small category; L is a class of diagrams in S , called limit diagrams; C is a class of diagrams in S , called colimit diagrams; σ is a function assigning to each diagram in L a cone and to each diagram in C a cocone. Definition B.0.2.
A sketch is • limit if C is empty; • colimit if L is empty; • mixed (used only in emphatic sense) if it’s not limit, nor colimit; • geometric if each cone is finite; • coherent if it is geometric and and every cocone is either finite or discrete,or it is a regular-epi specification . Definition B.0.3 (Morphism of Sketches) . Given two sketches S and T , a mor-phism of sketches f : S → T is a functor f : S → T mapping (co)limit diagramsinto (co)limits diagrams and proper (co)cones into (co)cones. Definition B.0.4 ( -category of Sketches) . The 2-category of sketches has sketchesas objects, morphism of sketches as 1-cells and natural transformations as 2-cells.
Definition B.0.5 (Category of models of a sketch) . For a sketch S and a (bi-complete) category C , the category Mod C ( S ) of C -models of the sketch is the fullsubcategory of C S of those functors that are models. If it’s not specified, by Mod ( S )we mean Mod
Set ( S ). Definition B.0.6 (Model of a sketch) . A model of a sketch S in a category C is afunctor f : S → C mapping each specified (co)cone to a (co)limit (co)cone. If it’snot specified a model is a Set -model.B.1.
Literature.
There exists a plethora of different and yet completely equivalentapproaches to the theory of sketches. We stick to the one that suits best our setting,following mainly [Bor94b][Chap. 5.6] or [AR94][Chap. 2.F]. Other authors, such as[MP89] and [Joh02b] use a different (and more classical) definition involving graphs.Sketches are normally used as generalized notion of theory. From this perspectivethese approaches are completely equivalent, because the underlying categories ofmodels are the same. [MP89][page 40] stresses that the graph-definition is a bitmore flexible in daily practice . Sketches were introduced by C. Ehresmann. Guitart,Lair and Burroni should definitely be mentioned among the relevant contributors.This list of references does not do justice to the French school, which has beenextremely prolific on this topic, yet, for the purpose of this paper the literatureabove will be more then sufficient. See [Joh02b][D2.1.2]. † B.2.
Sketches: logic and sketchable categories.
Sketches became quite com-mon among category theorists because of their expressiveness. In fact, they can beused as a categorical analog of those theories that can be axiomatized by (co)limitproperties. For example, in the previous section, essentially algebraic theories areprecisely those axiomatizable by finite limits.B.2.1.
From theories to sketches.
We have mentioned that a sketch can be seen asa kind of theory. This is much more than a motto, or a motivational presentation ofsketches. In fact, given a (infinitary) first order theory T , one can always constructin a more or less canonical way a sketch S T whose models are precisely the modelsof T . This is very well explained in [Joh02b][D2.2]; for the sake of exemplification,let us state the theorem which is most relevant to our context. Theorem B.2.1. If T is a (geometric) (coherent) theory, there there exists a(geometric) (coherent) sketch having the same category of models of T .Some readers might be unfamiliar with geometric and coherent theories; these arejust very specific fragments of first order (infinitary) logic. For a very detailed andclean treatment we suggest [Joh02b][D1.1]. Sketches are quite a handy notion oftheory because we can use morphisms of sketches as a notion of translation betweentheories. Proposition B.2.2 ([Bor94b][Ex. 5.7.14]) . If f : S → T is a morphism of sketches,then composition with f yields an (accessible) functor Mod ( S ) → Mod ( T ).B.2.2. Sketchability.
It should not be surprising that sketches can be used to ax-iomatize accessible and locally presentable categories too. The two following resultsappear, for example, in [AR94][2.F].
Theorem B.2.3.
A category is locally presentable if and only if it’s equivalent tothe category of models of a limit sketch.
Theorem B.2.4.
A category is accessible if and only if it’s equivalent to thecategory of models of a mixed sketch.
Appendix C. Topoi
Achtung! C.0.1.
In this section by topos we mean Grothendieck topos.
Definition C.0.2 (Topos) . A topos E is lex-reflective subcategory of a categoryof presheaves over a small category, i ∗ : Set C ◦ ⇆ E : i ∗ . Definition C.0.3 (Geometric morphism) . A geometric morphism of topoi f : E → F is an adjunction f ∗ : F ⇆ E : f ∗ whose left adjoint preserves finite limits (isleft exact). We will make extensive use of the following terminology: f ∗ is the inverse image functor; f ∗ is the direct image functor. Definition C.0.4 ( -category of Topoi) . The 2-category of topoi has topoi asobjects, geometric morphisms as 1-cells and natural transformations between leftadjoints as 2-cells. This means that it is a reflective subcategory and that the left adjoint preserves finite limits.Lex stands for left exact , and was originally motivated by homological algebra. Up to equivalence of categories. Notice that f ∗ is the right adjoint. ORMAL MODEL THEORY & HIGHER TOPOLOGY 21
C.1.
Literature.
There are several standard references for the theory of topoi. Tothe absolute beginner and even the experienced category theorist that does nothave much confidence with the topic, we recommend [Lei10]. Most of the technicalcontent of the paper can be understood via [LM94], a reference that we stronglysuggest to start and learn topos theory. Unfortunately, the approach of [LM94] isa bit different from ours, and even though its content is sufficient for this paper,the intuition that is provided is not 2-categorical enough for our purposes. Thereader might have to integrate with the encyclopedic [Joh02a, Joh02b]. A coupleof constructions that are quite relevant to us are contained only in [Bor94a], thatis otherwise very much equivalent to [LM94].C.2.
A comment on these definitions.
Topoi were defined by Grothendieck asa natural generalization of the category of sheaves Sh ( X ) over a topological space X . Their geometric nature was thus the first to be explored and exploited. Yet,with time, many other properties and facets of them have emerged, making themone of the main concepts in category theory between the 80’s and 90’s. Johnstone,in the preface of [Joh02a] gives 9 different interpretations of what a topos can be .In fact, this multi-faced nature of the concept of topos motivates the title of hisbook. In this paper we will concentrate on three main aspects of topos theory. • A topos is a (categorification of the concept of) locale; • A topos is a (family of Morita-equivalent) geometric theory; • A topos is an object in the 2-category of topoi.The first and the second aspects will be conceptual, and will allow us to inferqualitative results in geometry and logic, the last one will be our methodologicalpoint of view on topoi, and ultimately the main reason for which [LM94] might notbe a sufficient reference for this paper.C.3.
Site descriptions of topoi.
The first definition of topos that has been givenwas quite different from the one that we have introduced. As we have mentioned,topoi were introduced as category of sheaves over a space, thus the first definitionwas based on a generalization of this presentation. This is the theory of sites, andthe reader of [LM94] will recognize this approach in [LM94][Chap. 3]. In a nutshell,a site (
C, J ) is the data of a category C together a notion of covering families. Forexample, in the case of a topological space, C is the locale of open sets of X , and J is given by the open covers. Thus, a topos can be defined to be a category ofsheaves over a small site, E ≃ Sh ( C, J ) . Sh ( C, J ) is defined as a full subcategory of
Set C ◦ , which turn out to be lex-reflective.That’s the technical bridge between the site-theoretic description of a topos andthe one at the beginning of the section. Site theory is extremely useful in orderto study topoi as categories , while our approach is much more useful in order tostudy them as objects . We will never use explicitly site theory in the paper, withthe exception of a couple of proofs and a couple of examples.C.4. Topoi and Geometry.
It’s a bit hard to convey the relationship betweentopos theory and geometry in a short subsection. We mainly address the reader to[Lei10]. Let us just mention that to every topological space X , one can associateits category of sheaves Sh ( X ) (and this category is a topos), moreover, this assign-ment is a very strong topological invariant. For this reason, the study of Sh ( X )is equivalent to the study of X from the perspective of the topologist, and is veryconvenient in algebraic geometry and algebraic topology. For example, the categoryof sets is the topos of sheaves over the one-point-space, Set ∼ = Sh ( • ) † for this reason, category-theorists sometime call Set the point . This intuition isconsistent with the fact that
Set is the terminal object in the category of topoi.Moreover, as a point p ∈ X of a topological space X is a continuous function p : • → X , a point of a topos G is a geometric morphism p : Set → G . Parallelismsof this kind have motivated most of the definitions of topos theory and most have ledto results very similar to those that were achieved in formal topology (namely thetheory of locales). The class of points of a topos E has a structure of category pt ( E )in a natural way, the arrows being natural transformations between the inverseimages of the geometric morphisms.C.5. Topoi and Logic.
Geometric logic and topos theory are tightly bound to-gether. Indeed, for a geometric theory T it is possible to build a topos Set [ T ] (theclassifying topos of T ) whose category of points is precisely the category of modelsof T , Mod ( T ) ∼ = pt ( Set [ T ]) . This amounts to the theory of classifying topoi [LM94][Chap. X] and each toposclassifies a geometric theory. This gives us a logical interpretation of a topos. Eachtopos is a geometric theory , which in fact can be recovered from any of its sitesof definition. Obviously, for each site that describes the same topos we obtaina different theory. Yet, these theories have the same category of models (in anytopos). In this paper we will exploit the construction of [Bor94a] to show that toeach geometric sketch (a kind of theory), one can associate a topos whose pointsare precisely the models of the sketch. This is another way to say that the categoryof topoi can internalize a geometric logic.C.6.
Special classes of topoi.
In the paper we will study some relevant classesof topoi. In this subsection we recall all of them and give a good reference to checkfurther details. These references will be repeated in the relevant chapters.Topoi Referenceconnected [Joh02b][C1.5.7]compact [Joh02b][C3.2]atomic [Joh02b][C3.5]locally decidable [Joh02b][C5.4]coherent [Joh02b][D3.3]boolean [Joh02b][D3.4, D4.5], [Joh02a][A4.5.22]
Appendix D. Ionads
D.1.
Garner’s definitions.Definition D.1.1 (Ionad) . An ionad X = ( X, Int) is a set X together with acomonad Int : Set X → Set X preserving finite limits. Definition D.1.2 (Category of opens of a ionad) . The category of opens O ( X ) ofa ionad X = ( X, Int) is the category of coalgebras of Int. We shall denote by U X the forgetful functor U X : O ( X ) → Set X . Definition D.1.3 (Morphism of Ionads) . A morphism of ionads f : X → Y is acouple ( f, f ♯ ) where f : X → Y is a set function and f ♯ is a lift of f ∗ , O ( Y ) O ( X ) Set Y Set Xf ♯ U Y U X f ∗ ORMAL MODEL THEORY & HIGHER TOPOLOGY 23
Definition D.1.4 (Specialization of morphism of ionads) . Given two morphism ofionads f, g : X → Y , a specialization of morphism of ionads α : f ⇒ g is a naturaltransformation between f ♯ and g ♯ , O ( Y ) O ( X ) f ♯ g ♯ α Definition D.1.5 ( -category of Ionads) . The 2-category of ionads has ionads asobjects, morphism of ionads as 1-cells and specializations as 2-cells.
Definition D.1.6 (Bounded Ionads) . A ionad X is bounded if O ( X ) is a topos.D.2. Ionads and topological spaces.
Ionads were defined by Garner in [Gar12],and to our knowledge that’s all the literature available on the topic. His definitionis designed to generalize the definition of topological space. Indeed a topologicalspace X is the data of a set (of points) and an interior operator,Int : 2 X → X . Garner builds on the well known analogy between powerset and presheaf categoriesand extends the notion of interior operator to a presheaf category. The whole theoryis extremely consistent with the expectations: while the poset of (co)algebras forthe interior operator is the locale of open sets of a topological space, the category ofcoalgebras of a ionad is a topos, a natural categorification of the concept of locale.D.3.
A generalization and two related propositions.
In his paper Garnermentions that in giving the definition of ionad he could have chosen a categoryinstead of a set [Gar12][Rem. 2.4], let us quote his own comment on the definition.[[Gar12], Rem. 2.4] In the definition of ionad, we have chosen tohave a mere set of points, rather than a category of them. We do sofor a number of reasons. The first is that this choice mirrors mostclosely the definition of topological space, where we have a set, andnot a poset, of points. The second is that we would in fact obtain noextra generality by allowing a category of points. We may see thisanalogy with the topological case, where to give an interior operatoron a poset of points ( X, ≤ ) is equally well to give a topology O ( X )on X such that every open set is upwards-closed with respect to ≤ . Similarly, to equip a small category C with an interior comonadis equally well to give an interior comonad on X := ob C togetherwith a factorization of the forgetful functor O ( X ) → Set X throughthe presheaf category Set C ; this is an easy consequence of Example2.7 below. However, the most compelling reason for not admittinga category of points is that, if we were to do so, then adjunctionssuch as that between the category of ionads and the category oftopological spaces would no longer exist. Note that, although wedo not allow a category of points, the points of any (well-behaved)ionad bear nonetheless a canonical category structure – describedin Definition 5.7 and Remark 5.9 below – which may be understoodas a generalization of the specialization ordering on the points of aspace.We have decided to allow ionads over a category, even a locally small (but possiblylarge) one. We will need this definition later in the text to establish a connectionbetween ionads and topoi. While the structure of category is somewhat accessory,as Garner observes, the one of proper class will be absolutely needed. † Definition D.3.1 (Generalized Ionads) . A generalized ionad X = ( X, Int) is alocally small (but possibly large) pre-finitely cocomplete category X together witha lex comonad Int : ¶ ( X ) → ¶ ( X ). Achtung! D.3.2.
We will always omit the adjective generalized . Remark D.3.3.
We are well aware that the notion of generalized ionad seems abit puzzling at first sight.
Why isn’t it just the data of a locally small category X together with a lex comonad on Set X ? The answer to this question is a bitdelicate, having both a technical and a conceptual aspect. Let us first make precisethe notion above, introducing all the concepts that are mentioned, then we willdiscuss in what sense this is the correct notion of generalized ionad.
Remark D.3.4.
In a nutshell, ¶ ( X ) is a well-behaved full subcategory of Set X ,while the existence of finite pre-colimits will ensure us that ¶ ( X ) has finite limits.Let us dedicate some remarks to make these hints more precise. Remark D.3.5 (On small (co)presheaves) . By ¶ ( X ) we mean the full subcategoryof Set X made by small copresheaves over X , namely those functors X → Set thatare small colimits of corepresentables (in
Set X ). This is a locally small category,as opposed to Set X which might be locally large. The study of small presheaves X ◦ → Set over a category X is quite important with respect to the topic of freecompletions under limits and under colimits. Obviously, when X is small, everypresheaf is small. Given a category X , its category of small presheaves is usuallyindicated by P ( X ), while P ♯ ( X ) is P ( X ◦ ) ◦ . The most updated account on theproperty of P ( X ) is given by [AR18] and [DL07]. P ( X ) is the free completion of X under colimits, while P ♯ ( X ) is the free completion of X under limits. The followingequation clarifies the relationship between P , ¶ and P ♯ , P ♯ ( X ) ◦ = ¶ ( X ) = P ( X ◦ ) . This means that ¶ ( X ) is the free completion of X ◦ under colimits. Remark D.3.6.
The category of small presheaves P ( X ) over a (locally small)large category X is a bit pathological, especially if we keep the intuition that wehave when X is small. In full generality P ( X ) is not complete, nor it has any limitwhatsoever. Yet, under some smallness condition most of the relevant propertiesof P ( X ) remain true. Below we recall a good example of this behavior, and weaddress the reader to [AR18] for a for complete account. Proposition D.3.7 ([AR18][Cor. 3.8]) . P ( X ) is (finitely) complete if and only if X is (finitely) pre-complete . Corollary D.3.8. If X is finitely pre-cocomplete, then ¶ ( X ) has finite limits.A precise understanding of the notion of pre-cocomplete category is actually notneeded for our purposes, the following sufficient condition will be more than enoughthrough the paper. Corollary D.3.9 ([AR18][Exa. 3.5 (b) and (c)]) . If X is small or it is accessible,then ¶ ( X ) is complete.What must be understood is that being pre-complete, or pre-cocomplete should notbe seen as a completeness-like property, instead it is much more like a smallnessassumption. Example D.3.10 (Ionads are generalized ionads) . It is obvious from the previousdiscussion that a ionad is a generalized ionad. See [AR18][Def. 3.3].
ORMAL MODEL THEORY & HIGHER TOPOLOGY 25
Remark D.3.11 (Small copresheaves vs copresheaves) . When X is a finitely pre-cocomplete category, ¶ ( X ) is an infinitary pretopos and finite limits are nice in thesense that they can be computed in Set X . Being an infinitary pretopos, togetherwith being the free completion under (small) colimits makes the conceptual analogybetween ¶ ( X ) and 2 X nice and tight, but there is also a technical reason to prefersmall copresheaves to copresheaves. Proposition D.3.12. If f ∗ : G → ¶ ( X ) is a cocontinuous functor from a totalcategory, then it has a right adjoint f ∗ . Remark D.3.13.
The result above allows to produce comonads on ¶ ( X ) (justcompose f ∗ f ∗ ) and follows from the general theory of total categories, but needs ¶ ( X ) to be locally small to stay in place. Thus the choice of Set X would havegenerated size issues. A similar issue would arise with Kan extensions. Achtung! D.3.14. ¶ ( X ) is a (Grothendieck) topos if and only if X is an essentiallysmall category, thus in most of the examples of our interest ¶ ( X ) will not be aGrothendieck topos. Yet, we feel free to use a part of the terminology from topostheory (geometric morphism, geometric surjection, geometric embedding), becauseit is an infinitary pretopos (and thus only lacks a generator to be a topos). Remark D.3.15.
In analogy with the notion of base for a topology, Garner definesthe notion of base of a ionad [Gar12][Def. 3.1, Rem. 3.2]. This notion will be ahandy technical tool in the paper. Our definition is pretty much equivalent toGarner’s one (up to the fact that we keep flexibility on the size of the base) and isdesigned to be easier to handle in our setting.
Definition D.3.16 (Base of a ionad) . Let X = ( X, Int) be a ionad. We say thata flat functor e : B → ¶ ( X ) generates the ionad if Int is naturally isomorphic tothe density comonad of e , Int ∼ = lan e e. Example D.3.17.
The forgetful functor U X : O ( X ) → ¶ ( X ) is always a basis forthe ionad X . This follows from the basic theory about density comonads: when U X is a left adjoint, its density comonad coincides with the comonad induced by itsadjunction. This observation does not appear in [Gar12] because he only definedsmall bases, and it almost never happens that O ( X ) is a small category.In [Gar12][3.6, 3.7], the author lists three equivalent conditions for a ionad tobe bounded. The conceptual one is obviously that the category of opens is aGrothendieck topos, while the other ones are more or less technical. In our treat-ment the equivalence between the three conditions would be false. But we have thefollowing characterization. Proposition D.3.18.
A ionad X = ( X, Int) is bounded if any of the followingequivalent conditions is verified:(1) O ( X ) is a topos.(2) there exist a Grothendieck topos G and a geometric surjection f : ¶ ( X ) ։ G such that Int ∼ = f ∗ f ∗ .(3) there exist a Grothendieck topos G , a geometric surjection f : ¶ ( X ) ։ G and a flat functor e : B → G such that f ∗ e generates the ionad. Proof.
Clearly (1) implies (2). For the implication (2) ⇒ (3), it’s enough to choose e : B → G to be the inclusion of any generator of G . Let us discuss the implication(3) ⇒ (1). Let E be the category of coalgebras for the density comonad of e and This definition is just a bit different from Garner’s original definition [Gar12][Def. 3.1, Rem.3.2]. We stress that in this definition, we allow for large basis. † call g : G → E the geometric surjection induced by the comonad, (in particular lan e e ∼ = g ∗ g ∗ ). We claim that E ≃ O ( X ). Invoking [LM94][VII.4 Prop. 4] andbecause geometric surjections compose, we have E ≃ coAlg ( f ∗ g ∗ g ∗ f ∗ ). The thesisfollows from the observation thatInt ∼ = lan f ∗ e ( f ∗ e ) ∼ = lan f ∗ ( lan e ( f ∗ e )) ∼ = lan f ∗ ( f ∗ lan e e ) ∼ = f ∗ g ∗ g ∗ f ∗ . (cid:3) Remark D.3.19.
In the paper, we will need a practical way to induce morphismof ionads. The following proposition does not appear in [Gar12] and will be ourmain morphism generator . From the perspective of developing technical tool in thetheory of ionads, this proposition has an interest in its own right.
Remark D.3.20.
The proposition below categorifies a basic lemma in generaltopology: let f : X → Y be a function between topological spaces, and let B X and B Y be bases for the respective topologies. If f − ( B Y ) ⊂ B X , then f is continuous.Our original proof has been simplified by Richard Garner during the reviewingprocess of the author’s Ph.D thesis. Proposition D.3.21 (Generator of morphism of ionads) . Let X and Y be ionads,respectively generated by bases e X : B → ¶ ( X ) and e Y : C → ¶ ( Y ). Let f : X → Y a functor admitting a lift as in the diagram below. C B ¶ ( Y ) ¶ ( X ) e Y f ⋄ e X f ∗ If one of the two following conditions holds, then f induces a morphism of ionads( f, f ♯ ): Proof.
By the discussion in [Gar12][Exa. 4.6, diagram (6)], it is enough to providea morphism as described in the diagram below. C O ( lan e X e X ) ¶ ( Y ) ¶ ( X ) e Y f ′ U X f ∗ Also, [Gar12][Exa. 4.6] shows that giving a map of ionads X → Y is the same ofgiving f : X → Y and a lift of C → ¶ ( Y ) →¶ ( X ) through O ( X ). Applying thisto the identity map X → X we get a lift of B → ¶ ( X ) trough O ( lan e X e X ). Nowcomposing that with C → B gives the desired square. C B O ( lan e X e X ) ¶ ( Y ) ¶ ( X ) e Y f ⋄ f ′ e X U X f ∗ (cid:3) ORMAL MODEL THEORY & HIGHER TOPOLOGY 27
Acknowledgements
The content of the first section was substantially inspired by some private conver-sations with Jiˇr´ı Rosick´y. I am grateful to Simon Herny for some very constructivediscussions on Sec. 3 and 4. I am indebted to Axel Osmond for having read andcommented a preliminary draft of this paper.
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