aa r X i v : . [ m a t h . C T ] O c t GENERAL FACTS ON THE SCOTT ADJUNCTION
IVAN DI LIBERTI † Abstract.
We introduce, comment and develop the Scott adjunction, mostlyfrom the point of view of a category theorist. Besides its technical and concep-tual aspects, in a nutshell we provide a categorification of the Scott topologyover a posets with directed suprema. From a technical point of view we es-tablish an adjunction between accessible categories with directed colimits andGrothendieck topoi. We show that the bicategory of topoi is enriched overthe 2-category of accessible categories with directed colimits and it has tensorswith respect to this enrichment. The Scott adjunction (ri-)emerges naturallyfrom this observation.
Contents
1. Introduction 12. The Scott Adjunction 23. Category theory 104. Toolbox 15Acknowledgements 20References 201.
Introduction
In the 1980’s Scott’s work on dcpos has deeply impacted domain theory [AJ94],with motivations coming from theoretical computer sciences. Among his contribu-tions, he introduced the
Scott topology on a poset with directed colimits. Givensuch a poset, ( P, ≤ ) we can define a topology on P where opens are precisely thosesubsets whose characteristic function (from the given preorder into the preorder oftruth values) preserve directed joins. This construction amounts to a functor fromthe category of posets with directed colimits to the category of locales, S : Pos ω → Locwhich assigns to each poset with directed colimits, its frame of Scott opens. Sinceits introduction, the Scott topology on a poset has been studied in deep detail.Yet, the study of the functorial properties of this construction did not have muchluck. This becomes even more evident when we notice that the Scott constructionhappens to be the left adjoint of the functor of points pt : Loc → Pos ω , S : Pos ω ⇆ Loc : pt . Indeed the crude set of points of a locale admits a structure of poset with directedcolimits [Vic07], and the Scott construction offers a left adjoint for this assignement.Surprisingly, such observation has never appeared in the literature. † 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. † Our task is to categorify the Scott construction. From a technical point of viewwe establish a biadjunction between accessible categories with directed colimits andGrothendieck topoi, S : Acc ω ⇆ Topoi ω : pt . We study its main properties and try to unveil its ultimate (categorical) nature.Indeed, we show that the bicategory of topoi is enriched over accessible categorieswith directed colimits, and we rediscover the Scott adjunction as an instance ofbeing tensored. Together with [Lib20b], [Lib20a], this is one of three preprints whichaccounts on the content of the author’s Ph.D thesis. This preprint is devoted tosettle the categorical framework in which the theory is developed. The qualitativecontent of the adjunction is twofold. On one hand it has a very clean geometricinterpretation, whose roots belong to Stone-like dualities and Scott domains. Onthe other, it can be seen as a syntax-semantics duality between formal model theoryand geometric logic. Those points of view will be separately inspected in [Lib20b]and [Lib20a].This preprint has a predecessor, namely a joint collaboration of the author to-gether with Simon Henry [Hen19]. There, Henry uses the technology offered bythe Scott adjunction to solve an axiomatizabiliy question asked by Jiˇr´ı Rosick´y atthe international conference Category Theory 14. While Sec. 2 is just a more re-fined presentation of the results contained in [Hen19], the main contribution ofthis preprint are Sec. 3 and 4. Sec. 3 puts the Scott adjunction into perspective,recovering the adjunction from the fact that Topoi are enriched over accessible cat-egories. Sec. 4 is intended to be a collection of relevant and technical results onthe adjunction that are rather useful for the applications.1.1.
Organization of the paper.
Sec. 2 accounts on the Scott construction , mapping an accessible category withdirected colimits to its
Scott topos , A S ( A ) . This amounts to a functorwhich is left adjoint to taking the category of points of a topos E pt ( E ) . Sec. 3 makes a step back and studies some 2-categorical property of the 2-categoryof topoi, showing that Topoi is enriched over the 2-category of accessiblecategories with directed colimits and it has tensors with respect to thisenrichment. The Scott adjunction (ri-)emerges naturally from this obser-vation. The section also discusses the existence of certain cotensors.Sec. 4 accounts on a collection of technical properties of the functors S and pt ,such as the preservation of relevant notions of monomorphisms. Among theother results, the section introduces and studies the notion of topologicalembedding of accessible categories with directed colimits.2. The Scott Adjunction
In this section we provide enough information to understand the crude statementof the adjunction and we touch on these contextualizations. One could say that thissection, together with a couple of results that appear in the Toolbox, is a report ofour collaboration with Simon Henry [Hen19].
Structure.
The exposition is organized as follows:Sec. 2.1 we introduce the constructions involved in the Scott adjunction;Sec. 2.2 we provide some comments and insights on the first section;Sec. 2.3 we give a quick generalization of the adjunction and discuss its interactionwith the standard theorem;Sec. 2.4 we prove the Scott adjunction.
ENERAL FACTS ON THE SCOTT ADJUNCTION 3
The Scott adjunction: definitions and constructions.
We begin by giv-ing the crude statement of the adjunction, then we proceed to construct and describeall the objects involved in the theorem. The actual proof of Thm. 2.1 will close thesection.
Theorem 2.1 ([Hen19][Prop. 2.3] The Scott adjunction) . The 2-functor of points pt : Topoi → Acc ω has a left biadjoint S , yielding the Scott biadjunction, S : Acc ω ⇆ Topoi : pt . Remark 2.2 (Characters on the stage) . Acc ω is the 2-category of accessible cat-egories with directed colimits, a 1-cell is a functor preserving directed colimits,2-cells are natural transformations. Topoi is the 2-category of Grothendieck topoi.A 1-cell is a geometric morphism and has the direction of the right adjoint. 2-cellsare natural transformation between left adjoints. Remark 2.3 ( -categorical warnings) . Both Acc ω and Topoi are 2-categories, butmost of the time our intuition and our treatment of them will be 1-categorical,we will essentially downgrade the adjunction to a 1-adjunction where everythingworks up to equivalence of categories . We feel free to use any classical result about1-adjunction, paying the price of decorating any statement with the correct use ofthe word pseudo . For example, right adjoints preserve pseudo-limits, and pseudo-monomorphisms. Remark 2.4 (The functor pt ) . The functor of points pt belongs to the literaturesince quite some time, pt is the covariant hom functor Topoi( Set , − ). It maps aGrothendieck 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.2.1.1. The Scott construction.
Construction 2.5 (The Scott construction) . We recall the construction of S from[Hen19]. Let A be an accessible category with directed colimits. S ( A ) is defined asthe category the category of functors preserving directed colimits into sets. S ( A ) = Acc ω ( A , Set ) . For a functor f : A → B be a 1-cell in Acc ω , the geometric morphism S f is definedby precomposition as described below. A S AB S B f f ∗ f ∗ ⊣ S f = ( f ∗ ⊣ f ∗ ) is defined as follows: f ∗ is the precomposition functor f ∗ ( g ) = g ◦ f .This is well defined because f preserve directed colimits. f ∗ is a functor preservingall colimits between locally presentable categories and thus has a right adjoint bythe adjoint functor theorem , that we indicate with f ∗ . Observe that f ∗ preservesfinite limits because finite limits commute with directed colimits in Set . This is shown in 2.6. Apply the dual of [Bor94b][Thm. 3.3.4] in combination with [AR94][Thm 1.58].
IVAN DI LIBERTI † Remark 2.6 ( S ( A ) is a topos) . Together with 2.5 this shows that the Scott con-struction provides a 2-functor S : Acc ω → Topoi. A proof has already appearedin [Hen19][2.2] with a practically identical idea. The proof relies on the fact that,since finite limits commute with directed colimits, the category S ( A ) inherits fromits inclusion in the category of all functors A → Set all the relevant exactnesscondition prescribed by Giraud axioms. The rest of the proof is devoted to providea generator for S ( A ). In the proof below we pack in categorical technology theproof-line above. Proof of 2.6.
By definition A must be λ -accessible for some λ . Obviously Acc ω ( A , Set )sits inside λ -Acc( A , Set ). Recall that λ -Acc( A , Set ) is equivalent to
Set A λ by therestriction-Kan extension paradigm and the universal property of Ind λ -completion.This inclusion i : Acc ω ( A , Set ) ֒ → Set A λ , preserves all colimits and finite limits,this is easy to show and depends on one hand on how colimits are computed inthis category of functors, and on the other hand on the fact that in Set directedcolimits commute with finite limits. By the adjoint functor theorem, Acc ω ( A , Set )amounts to a coreflective subcategory of a topos whose associated comonad is leftexact. By [LM94][V.8 Thm.4], it is a topos. (cid:3)
Remark 2.7 (A description of f ∗ ) . In order to have a better understanding ofthe right adjoint f ∗ , which in the previous remark was shown to exist via a specialversion of the adjoint functor theorem, we shall fit the adjunction ( f ∗ ⊣ f ∗ ) into amore general picture. We start by introducing the diagram below. S A S B ¶ ( A ) ¶ ( B ) ι A f ∗ ι B f ∗ ran f lan f f ∗ (1) By ¶ ( A ) we mean the category of small copresheaves over A . Observe thatthe natural inclusion ι A of S A in ¶ ( A ) has a right adjoint r A , namely S A iscoreflective and it coincides with the algebras for the comonad ι A ◦ r A . If weignore the evident size issue for which ¶ ( A ) is not a topos, the adjunction ι A ⊣ r A amounts to a geometric surjection r : ¶ ( A ) → S A .(2) The left adjoint lan f to f ∗ does exist because f preserve directed colimits,while in principle ran f may not exists because it is not possible to cut downthe size of the limit in the ran-limit-formula. Yet, for those functors forwhich it is defined, it provides a right adjoint for f ∗ . Observe that sincethe f ∗ on the top is the restriction of the f ∗ on the bottom, and ι A , B arefully faithful, f ∗ has to match with r B ◦ ran f ◦ ι A , when this composition iswell defined, f ∗ ∼ = r B ◦ ran f ◦ ι A , indeed the left adjoint f ∗ on the top coincides with f ∗ ◦ ι B and by uniquenessof the right adjoint one obtains precisely the equation above. Later in thetext this formula will prove to be useful. We can already use it to havesome intuition on the behavior f ∗ , indeed f ∗ ( p ) is the best approximationof ran f ( p ) preserving directed colimits. In particular if it happens for somereason that ran f ( p ) preserves directed colimits, then this is precisely thevalue of f ∗ ( p ). This will be shown in the remark below.
ENERAL FACTS ON THE SCOTT ADJUNCTION 5
Remark 2.8 ( S ( A ) is coreflective in ¶ ( A ) ) . We would have liked to have a one-line-motivation of the fact that the inclusion i A : S ( A ) → ¶ ( A ) has a right adjoint r A , unfortunately this result is true for a rather technical argument. By a generalresult of Kelly, i A has a right adjoint if and only if lan i A (1 S ( A ) ) exists and i A preservesit. Since S ( A ) is small cocomplete, if lan i A (1 S ( A ) ) exists, it must be pointwise andthus i will preserve it because it is a cocontinuous functor. Thus it is enough toprove that lan i A (1 S ( A ) ) exists. Anyone would be tempted to apply [Bor94b][3.7.2],unfortunately S ( A ) is not a small category. In order to cut down this size issue, weuse the fact that S ( A ) is a topos and thus have a dense generator j : G → S ( A ).Now, we observe that lan i A (1 S ( A ) ) = lan i A ( lan j ( j )) = lan i A ◦ j j. Finally, on the latter left Kan extension we can apply [Bor94b][3.7.2], because G isa small category. Remark 2.9.
Let A be a λ -accessible category, then S ( A ) can be described as thefull subcategory of Set A λ of those functors preserving λ -small ω -filtered colimits.A proof of this observation can be found in [Hen19][2.2], and in fact shows that S ( A ) has a generator.2.2. Comments and suggestions.Remark 2.10 (Cameos in the existing literature) . Despite the name, neither theadjunction nor the construction is due to Scott and was presented for the first timein [Hen19]. It implicitly appears in special cases both in the classical literature[Joh02b] and in some recent developments [AL18]. Karazeris introduces the notionof Scott topos of a finitely accessible category K in [Kar01], this notion coincideswith S ( K ), as the name suggests. In [Lib20b] we will make the connection withsome seminal works of Scott and clarify the reason for which this is the correctcategorification of a construction which is due to him. As observed in [Hen19][2.4],the Scott construction is the categorification of the usual Scott topology on a posetwith directed joins. This will help us to develop a geometric intuition on accessiblecategories with directed colimits; they will look like the underlying set of sometopological space. We cannot say to be satisfied with this choice of name for theadjunction, but we did not manage to come up with a better name. Remark 2.11 (The duality-pattern) . A duality-pattern is an adjunction that iscontravariantly induced by a dualizing object. For example, the famous dual ad-junction between frames and topological spaces [LM94][Chap. IX], O : Top ⇆ Frm ◦ : pt is induced by the Sierpinski space T . Indeed, since it admits a natural structureof frame, and a natural structure of topological space the adjunction above can berecovered in the form Top ( − , T ) : Top ⇆ Frm ◦ : Frm ( − , T ) . Most of the known topological dualities are induced in this way. The interestedreader might want to consult [PT91]. Makkai has shown ([MP87], [Mak88]) thatrelevant families of syntax-semantics dualities can be induced in this way using thecategory of sets as a dualizing object. In this fashion, the content of Rem. 2.5together with Rem. 2.4 acknowledges that S ⊣ pt is essentially induced by theobject Set that inhabits both the 2-categories.
Remark 2.12 (Generalized axiomatizations) . As was suggested by Joyal, the cat-egory Logoi = Topoi ◦ can be seen as the category of geometric theories. Caramello IVAN DI LIBERTI † [Car10] pushes on the same idea stressing the fact that a topos is a Morita-equivalence class of geometric theories. In this perspective the Scott adjunction,which in this case is a dual adjunctionAcc ω ⇆ Logoi ◦ , presents S ( A ) as a free geometric theory attached to the accessible category A that is willing to axiomatize A . When A has a faithful functor preserving directedcolimits into the category of sets, S ( A ) axiomatizes an envelope of A , as provedin 4.15. A logical understanding of the adjunction will be developed in [Lib20a],where we connect the Scott adjunction to the theory of classifying topoi and to theseminal works of Lawvere and Linton in categorical logic. This intuition will beused also to give a topos theoretic approach to abstract elementary classes. Remark 2.13 (Trivial behaviors and Diaconescu) . If K is a finitely accessiblecategory, S ( K ) coincides with the presheaf topos Set K ω , where we indicated with K ω the full subcategory of finitely presentable objects. This follows directly fromthe following chain of equivalences, S ( K ) = Acc ω ( K , Set ) ≃ Acc ω ( Ind ( K ω ) , Set ) ≃ Set K ω . As a consequence of Diaconescu theorem [Joh02a][B3.2.7] and the characterizationof the
Ind -completion via flat functors, when restricted to finitely accessible cate-gories, the Scott adjunction yields a biequivalence of 2-categories ω -Acc ≃ Presh,with Presh the full 2-subcategory of presheaf topoi.Acc ω Topoi ω -Acc Presh Spt
This observation is not new to literature, the proof of [Joh02b][C4.3.6] gives thisspecial case of the Scott adjunction. It is very surprising that the book does notinvestigate, or even mention the existence of the Scott adjunction, since it gets veryclose to defining it explicitly.
Theorem 2.14.
The Scott adjunction restricts to a biequivalence of 2-categoriesbetween the 2-category of finitely accessible categories and the 2-category of presheaftopoi . S : ω -Acc ⇆ Presh : pt . Proof.
The previous remark has shown that when A is finitely accessible, S ( A ) isa presheaf topos and that, when E is a presheaf topos, pt ( E ) is finitely accessible.To finish, we show that in this case the unit and the counit of the Scott adjunctionare equivalence of categories. This is in fact essentially shown by the previousconsiderations. ( ptS )( Ind ( C )) ≃ pt ( Set C ) Diac ≃ Ind ( C ) . ( Spt )( Set C ) Diac ≃ S ( Ind ( C )) ≃ Set C . Notice that the equivalences above are precisely the unit and the counit of the Scottadjunction as described in Sec 2.4 of this Chapter. (cid:3) With finitely accessible functors and natural transformation. With geometric morphisms and natural transformations between left adjoints.
ENERAL FACTS ON THE SCOTT ADJUNCTION 7
Remark 2.15.
Thus, the Scott adjunction must induce an equivalence of categoriesbetween the Cauchy completions of ω -Acc and Presh. The Cauchy completion of ω -Acc is the full subcategory of Acc ω of continuous categories [JJ82]. Continu-ous categories are the categorification of the notion of continuous poset and canbe characterized as split subobjects of finitely accessible categories in Acc ω . In[Joh02b][C4.3.6] Johnstone observe that if a continuous category is cocomplete,then the corresponding Scott topos is injective (with respect to geometric embed-dings) and vice-versa. Example 2.16.
As a direct consequence of Rem. 2.13, we can calculate the Scotttopos of
Set . S ( Set ) is
Set
FinSet . This topos is very often indicated as
Set [ O ], beingthe classifying topos of the theory of objects, i.e. the functor: Topoi( − , Set [ O ]) :Topoi ◦ → Cat coincides with the forgetful functor. As a reminder for the reader,we state clearly the equivalence: S ( Set ) ≃ Set [ O ] . Remark 2.17 (The Scott adjunction is not a biequivalence: Fields) . Whether theScott adjunction is a biequivalence is a very natural question to ask. Up to thispoint we noticed that on the subcategory of topoi of presheaf type the counit of theadjunction is in fact an equivalence of categories. Since presheaf topoi are a quiterelevant class of topoi one might think that the whole bi-adjunction amounts to abiequivalence. That’s not the case: in this remark we provide a topos F such thatthe counit ǫ F : Spt F → F is not an equivalence of categories. Let F be the classifying topos of the theory ofgeometric fields [Joh02b][D3.1.11(b)]. Its category of points is the category of fields Fld , since this category is finitely accessible the Scott topos
Spt ( F ) is of presheaftype by Rem. 2.13, Spt ( F ) = S ( Fld ) 2 . ∼ = Set
Fld ω . It was shown in [Bek04][Cor 2.2] that F cannot be of presheaf type, and thus ǫ F cannot be an equivalence of categories. Remark 2.18 (Classifying topoi for categories of diagrams) . Let us give a proof inour framework of a well known fact in topos theory, namely that a the category ofdiagrams over the category of points of a topos can be axiomatized by a geometrictheory. This means that there exists a topos F such that pt ( E ) C ≃ pt ( F ) . The proof follows from the following chain of equivalences. pt ( E ) C =Cat( C, pt ( E )) ≃ Acc ω ( Ind ( C ) , pt ( E )) ≃ Topoi(
SInd ( C ) , E ) ≃ Topoi(
Set C , E ) ≃ Topoi(
Set × Set C , E ) ≃ Topoi(
Set , E Set C ) ≃ pt ( E Set C ) . The free completions that adds splittings of pseudo -idempotents.
IVAN DI LIBERTI † The κ -Scott adjunction. The most natural generalization of the Scott ad-junction is the one in which directed colimits are replaced with κ -filtered colimitsand finite limits ( ω -small) are replaced with κ -small limits. This unveils the deepestreason for which the Scott adjunction exists: namely κ -directed colimits commutewith κ -small limits in the category of sets. Theorem 2.19. [Hen19][Prop 3.4] There is an 2-adjunction S κ : Acc κ ⇆ κ -Topoi : pt κ . Remark 2.20.
Acc κ is the 2-category of accessible categories with κ -directed col-imits, a 1-cell is a functor preserving κ -filtered colimits, 2-cells are natural trans-formations. Topoi κ is the 2-category of Groethendieck κ -topoi. A 1-cell is a κ -geometric morphism and has the direction of the right adjoint. 2-cells are natu-ral transformation between left adjoints. A κ -topos is a κ -exact localization of apresheaf category. These creatures are not completely new to the literature but theyappear sporadically and a systematic study is still missing. We should reference,though, the works of Espindola [Esp19]. We briefly recall the relevant definitions. Definition 2.21. A κ -topos is a κ -exact localization of a presheaf category. Definition 2.22. A κ -geometric morphism f : E → F between κ -topoi is anadjunction f ∗ : F ⇆ E : f ∗ whose left adjoint preserve κ -small limits. Remark 2.23.
It is quite evident that every remark until this point finds its direct κ -generalization substituting every occurrence of directed colimits with κ -directedcolimits. Remark 2.24.
Let A be a category in Acc ω . For a large enough κ the Scottadjunction axiomatizes A (in the sense of Rem. 2.12), in fact if A is κ -accessible pt κ S κ A ∼ = A , for the κ -version of Diaconescu Theorem, that in this text appears inRem. 2.13. Remark 2.25.
It pretty evident that λ -Topoi is a locally fully faithful sub 2-category of κ -Topoi when λ ≥ κ . The same is true for Acc λ and Acc κ . Thisobservation leads to a filtration of the categories Topoi and Acc ω as shown in thefollowing diagram, κ Acc κ κ -Topoi λ Acc λ λ -Topoi ω Acc ω ( ω -)Topoi ι λκ S κ i λκ pt κ ι ωλ S λ i ωλ pt λ Spt
Remark 2.26 (The diagram does not commute) . We depicted the previous dia-gram in order to trigger the reader’s pattern recognition and conjecture its com-mutativity. In this remark we stress that the diagram does not commute, meaningthat S λ ◦ ι λκ i λκ ◦ S κ , at least not in general. In fact, once the definitions are spelled out, there is abso-lutely no reasons why one should have commutativity in general. The same is truefor the right adjoint pt . ENERAL FACTS ON THE SCOTT ADJUNCTION 9
Remark 2.27.
In the following diagram we show the interaction between Rem.2.24 and the previous remark. Recall that presheaf categories belong to κ -topoi forevery κ . κ -Acc Acc κ κ -Topoi Presh λ -Acc Acc λ λ -Topoi Presh ω -Acc Acc ω ( ω -)Topoi Presh ι λκ S κ i λκ pt κ ι ωλ S λ i ωλ pt λ Spt
Remark 2.28.
It might be natural to conjecture that Presh happens to be T κ κ -Topoi.Simon Henry, has provided a counterexample to this superficial conjecture, namely Sh ([0 , Proof of the Scott Adjunction.
We end this section including a full proofof the Scott adjunction.
Proof of Thm. 2.1.
We prove that there exists an equivalence of categories,Topoi( S ( A ) , F ) ∼ = Acc ω ( A , pt ( F )) . The proof makes this equivalence evidently natural. This proof strategy is similarto that appearing in [Hen19], even thought it might look different at first sight.Topoi( S ( A ) , F ) ∼ =Cocontlex( F , S ( A )) ∼ =Cocontlex( F , Acc ω ( A , Set )) ∼ =Cat cocontlex,acc ω ( F × A , Set ) ∼ =Acc ω ( A , Cocontlex( F , Set )) ∼ =Acc ω ( A , Topoi(
Set , F )) . ∼ =Acc ω ( A , pt ( F )) . (cid:3) A description of the (co)unit.
We spell out the unit and the counit of the adjunc-tion. η For an accessible category with directed colimits A we must provide a func-tor η A : A → ptS ( A ). Define, η A ( a )( − ) := ( − )( a ) .η A ( a ) : S ( A ) → Set defined in this way is a functor preserving colimits andfinite limits and thus defines a point of S ( A ). ǫ The idea is very similar, for a topos E , we must provide a geometric mor-phism ǫ E : Spt ( E ) → E . Being a geometric morphism, it’s equivalent toprovide a cocontinuous and finite limits preserving functor ǫ ∗ E : E → Spt ( E ).Define, ǫ ∗ E ( e )( − ) = ( − ) ∗ ( e ) . (cid:3) † Category theory
This section is dedicated to a 2-categorical perspective on the Scott adjunctionand its main characters. We provide an overview of the categorical properties ofAcc ω and Topoi. Mainly, we show that the 2-category of topoi is enriched over Acc ω and has copowers. We show that this observation generalizes the Scott adjunctionin a precise sense. We discuss the 2-categorical properties of both the 2-categories,but this work is not original. We will provide references within the discussion.3.1. ω . (co)Limits in Acc ω . The literature contains a variety of results on the 2-dimensional structure of the 2-category Acc of accessible categories and accessiblefunctors. Among these, one should mention [MP89] for lax and pseudo-limits inAcc and [PR13] for colimits. Our main object of study, namely Acc ω , is a (non-full) subcategory of Acc, and thus it is a bit tricky to infer its properties from theexisting literature. Most of the work was successfully accomplished in [LR15]. Letus list the main results of these references that are related to Acc ω . Proposition 3.1 ([LR15][2.2]) . Acc ω is closed under pie-limits in Acc (and thusin the illegitimate 2-category of locally small categories). Proposition 3.2 (Slight refinement of [PR13][2.1]) . Every directed diagram ofaccessible categories and full embeddings preserving directed colimits has colimitin Cat, and is in fact the colimit in Acc ω .3.1.2. Acc ω is monoidal closed. This subsection discusses a monoidal closed struc-ture on Acc ω . The reader should keep in mind the monoidal product of modulesover a ring, because the construction is similar in spirit, at the end of the subsec-tion we will provide an hopefully convincing argument in order to show that theconstruction is similar for a quite quantitative reason. The main result of the sec-tion should be seen as a slight variation of [Kel82][6.5] where the enrichment baseis obviously the category of Sets and F -cocontinuity is replaces by preservation ofdirected colimits. Our result doesn’t technically follows from Kelly’s one becauseof size issues, but the general idea of the proof is in that spirit. Moreover, wefound it clearer to provide an explicit construction of the tensor product in ourspecific case. The reader is encouraged to check [hdl], where Brandenburg providesa concise presentations of Kelly’s construction. For a treatment of how the bilineartensor product on categories with certain colimits gives a monoidal bicategory werefer to [Bou17, HP02, L´o11]. Remark 3.3 (A natural internal hom) . Given two accessible categories A , B inAcc ω , the category of functors preserving directed colimits Acc ω ( A , B ) has directedcolimits and they are computed pointwise. Moreover it is easy to show that it issketchable and thus accessible. Indeed Acc ω ( A , B ) is accessibly embedded in B A λ and coincides with those functors A λ → B preserving λ -small directed colimits,which makes it clearly sketchable. Thus we obtain a 2-functor,[ − , − ] : Acc ◦ ω × Acc ω → Acc ω . In our analogy, this corresponds to the fact that the set of morphisms between twomodules over a ring
Mod ( M, N ) has a (pointwise) structure of module.
Remark 3.4 (Looking for a tensor product: the universal property) . Assume fora moment that the tensorial structure that we are looking for exists, then we wouldobtain a family of (pseudo)natural equivalences of categories,Acc ω ( A ⊗ B , C ) ≃ Acc ω ( A , [ B , C ]) ≃ ω -Bicocont( A × B , C ) . These are those limits can be reduced to products, inserters and equifiers.
ENERAL FACTS ON THE SCOTT ADJUNCTION 11
In the display we wrote ω -Bicocont( A × B , C ) to mean the category of those functorsthat preserve directed colimits in each variable. The equation gives us the universalproperty that should define A ⊗ B up to equivalence of categories and is consistentwith our ongoing analogy of modules over a ring, indeed the tensor product classifies bilinear maps . Construction 3.5 (Looking for a tensor product: the construction) . Let よ : A × B → P ( A × B ) be the Yoneda embedding of A × B corestricted to thefull subcategory of small presheaves [AR18]. Let B ( A , B ) be the full subcategoryof P ( A × B ) of those functors that preserve cofiltered limits in both variables .It is easy to show that B ( A , B ) is sketched by a limit theory, and thus is locallypresentable. The inclusion i : B ( A , B ) ֒ → P ( A × B ) defines a small-orthogonalityclass in P ( A × B ) and is thus reflective [AR94][1.37, 1.38]. Let L be the left adjointof the inclusion, as a result we obtain an adjunction, L : P ( A × B ) ⇆ B ( A , B ) : i. Now define A ⊗ B to be the smallest full subcategory of B ( A , B ) closed underdirected colimits and containing the image of L ◦ よ . Thm. [Kel82][6.23] in Kellyensures that A ⊗ B has the universal property described in Rem. 3.4 and thus isour tensor product. It might be a bit hard to see but this construction still followsour analogy, the tensor product of two modules is indeed built from free module onthe product and the bilinear relations. Theorem 3.6.
Acc ω , together with the tensor product ⊗ defined in Con. 3.5 andthe internal hom defined in Rem. 3.3 is a monoidal biclosed bicategory in thesense that there is are pseudo-equivalences of categoriesAcc ω ( A ⊗ B , C ) ≃ Acc ω ( A , [ B , C ]) , which are natural in C . Proof.
Follows directly from the discussion above. (cid:3)
Remark 3.7 (Up to iso/up to equivalence) . As in [Kel82][6.5], we will not distin-guish between the properties of this monoidal structure (where everything is trueup to equivalence of categories) and a usual one, where everything is true up toisomorphism. In our study this distinction never plays a rˆole, thus we will use theusual terminology about monoidal structures.
Remark 3.8 (The unit) . The unit of the above-mentioned monoidal structure isthe terminal category in Cat, which is also terminal in Acc ω . Remark 3.9 (Looking for a tensor product: an abstract overview) . In this subsec-tion we have used the case of modules over a ring as a kind of analogy/motivatingexample. In this remark we shall convince the reader that the analogy can bepushed much further. Let’s start by the observation that R - Mod is the categoryof algebras for the monad R [ − ] : Set → Set . The monoidal closed structure of
Mod can be recovered from the one of the category of sets (
Set , , × , [ − , − ]) via aclassical theorem proved by Kock in the seventies. It would not make the tractationmore readable to cite all the papers that are involved in this story, thus we mentionthe PhD thesis of Brandenburg [Bra14][Chap. 6] which provides a very coherentand elegant presentation of the literature. i.e., send filtered colimits in A or B to limits in Set . Here we are using that A and B are accessible in order to cut down the size of the orthogonality. This is not strictly true, because the definition of monoidal closed category does not allowfor equivalence of categories. We did not find a precise terminology in the literature and we feltnon-useful to introduce a new concept for such a small discrepancy. † Theorem 3.10. (Seal, 6.5.1 in [Bra14]) Let T be a coherent (symmetric) monoidalmonad on a (symmetric) monoidal category C with reflexive coequalizers. Then Mod ( T ) becomes a (symmetric) monoidal category.Now similarly to Mod ( R ) the 2-category of categories with directed colimits andfunctors preserving them is the category of (pseudo)algebras for the KZ monad ofthe Ind-completion over locally small categories Ind : Cat → Cat . [Bou17][6.7] provides a version of Seal’s theorem for monads over Cat. While it’squite easy to show that the completion under directed colimits meets many ofBourke’s hypotheses, we do not believe that it meets all of them, thus we did notmanage to apply a Kock-like result to derive Thm. 3.6. Yet, we think we haveprovided enough evidence that the analogy is not just motivational.3.2. (co)Limits in Topoi. The 2-categorical properties of the category of topoihave been studied in detail in the literature. We mention [Joh02b][B3.4] and [Lur09]as a main reference.3.2.2.
Enrichment over Acc ω , tensor and powers. The main content of this sub-subsection will be to show that the category of topoi and geometric morphisms(in the direction of the right adjoint) is enriched over Acc ω . Notice that formallyspeaking, we are enriching over a monoidal bicategory, thus the usual theory ofenrichment is not sufficient. As Garner pointed out to us, the theory of bicategoriesenriched in a monoidal bicategory is originally due to Bozapalides in the 1970s,though he was working without the appropriate technical notion; more precisedefinitions are in the PhD theses of Camordy and Lack; and everything is workedout in excruciating detail in [GS16]. Remark 3.11.
Recall that to provide such an enrichment means to(1) show that given two topoi E , F , the set of geometric morphisms Topoi( E , F )admits a structure of accessible category with directed colimits.(2) provide, for each triple of topoi E , F , G , a functor preserving directed col-imits ◦ : Topoi( E , F ) ⊗ Topoi( F , G ) → Topoi( E , G ) , making the relevant diagrams commute.(1) will be shown in Prop. 3.12, while (2) will be shown in Prop. 3.13. Proposition 3.12.
Let E , F be two topoi. Then the category of geometric mor-phisms Cocontlex( E , F ), whose objects are cocontinuous left exact functors andmorphisms are natural transformations is an accessible category with directed col-imits. Proof.
The proof goes as in [Bor94a][Cor.4.3.2],
Set plays no rˆole in the proof.What matters is that finite limits commute with directed colimits in a topos. (cid:3)
Proposition 3.13.
For each triple of topoi E , F , G , there exists a functor preserv-ing directed colimits ◦ : Topoi( E , F ) ⊗ Topoi( F , G ) → Topoi( E , G ) , making the relevant diagrams commute. ENERAL FACTS ON THE SCOTT ADJUNCTION 13
Proof.
We will only provide the composition. The relevant diagrams commutetrivially from the presentation of the composition. Recall that by 3.4 a map of theform ◦ : Topoi( E , F ) ⊗ Topoi( F , G ) → Topoi( E , G ) , preserving directed colimits isthe same of a functor ◦ : Topoi( E , F ) × Topoi( F , G ) → Topoi( E , G )preserving directed colimits in each variables. Obviously, since left adjoints canbe composed in Cat, we already have such a composition. It’s enough to showthat it preserves directed colimits in each variable. Indeed this is the case, becausedirected colimits in these categories are computed pointwise. (cid:3) Theorem 3.14.
The category of topoi is enriched over Acc ω . Proof.
Trivial from the previous discussion. (cid:3)
Tensors.
In this subsection we show that the 2-category of topoi has tensors(copowers) with respect to the enrichment of the previous section.
Remark 3.15.
Let us recall what means to have tensors for a category K enrichedover sets (that is, just a locally small category). To have tensors in this case meansthat we can define a functor ⊠ : Set × K → K in such a way that, K ( S ⊠ k, h ) ∼ = Set ( S, K ( k, h )) . For example, the category of modules over a ring has tensors given by the formula S ⊠ M := ⊕ S M ; indeed it is straightforward to observe that R - Mod ( ⊕ S M, N ) ∼ = Set ( S, R - Mod ( M, N )) . In this case, this follows from the universal property of the coproduct.
Remark 3.16 (The construction of tensors) . We shall define a 2-functor ⊠ : Acc ω × Topoi → Topoi . Our construction is reminiscent of the Scott adjunction, and wewill see that there is an extremely tight connection between the two. Given a topos E and an accessible category with directed colimits A we define, A ⊠ E := Acc ω ( A , E ) . In order to make this construction meaningful we need to accomplish two tasks:(1) show that the construction is well defined (on the level of objects), that is,show that Acc ω ( A , E ) is a topos.(2) describe the action of ⊠ on functors.We split these two tasks into two different remarks. Remark 3.17 (Acc ω ( A , E ) is a topos) . By definition A must be λ -accessible forsome λ . Obviously Acc ω ( A , E ) sits inside λ -Acc( A , E ). Recall that λ -Acc( A , E )is equivalent to E A λ by the restriction-Kan extension paradigm and the universalproperty of Ind λ -completion. The inclusion i : Acc ω ( A , E ) ֒ → E A λ , preserves allcolimits and finite limits, this is easy to show and depends on the one hand on howcolimits are computed in this category of functors, and on the other hand on thefact that in a topos directed colimits commute with finite limits. Thus Acc ω ( A , E )amounts to a coreflective subcategory of a topos whose associated comonad is leftexact. So by [LM94][V.8 Thm.4], it is a topos. Remark 3.18 (Action of ⊠ on functors) . Let f : A → E be in Acc ω ( A , E ) and let g : E → F be a geometric morphism. We must define a geometric morphism f ⊠ g : Acc ω ( A , E ) → Acc ω ( B , F ) . We shall describe the left adjoint ( f ⊠ g ) ∗ (which goes in the opposite direction( f ⊠ g ) ∗ : Acc ω ( B , F ) → Acc ω ( A , E )) by the following equation:( f ⊠ g ) ∗ ( s ) = g ∗ ◦ s ◦ f. † Proposition 3.19.
Topoi has tensors over Acc ω . Proof.
Putting together the content of 3.16, 3.17 and 3.18, we only need to showthat ⊠ has the correct universal property, that is:Topoi( A ⊠ E , F ) ∼ = Acc ω ( A , Topoi( E , F )) . When we spell out the actual meaning of the equation above, we discover that wedid all the relevant job in the previous remarks. Indeed the biggest obstruction wasthe well-posedness of the definition.Topoi( A ⊠ E , F ) ∼ =Cocontlex( F , A ⊠ E ) ∼ =Cocontlex( F , Acc ω ( A , E )) ∼ =Cat cocontlex,acc ω ( F × A , E ) ∼ =Acc ω ( A , Cocontlex( F , E )) ∼ =Acc ω ( A , Topoi( E , F )) . (cid:3) The Scott adjunction revisited.Remark 3.20 (Yet another proof of the Scott adjunction) . Let us start by men-tioning that we can re-obtain the Scott adjunction directly from the fact that Topoiis tensored over Acc ω . Indeed if we evaluate the equation in 3.19 when E is theterminal topos Set,Topoi( A ⊠ Set , F ) ∼ = Acc ω ( A , Topoi(
Set , F ))we obtain precisely the statement of the Scott adjunction,Topoi( S ( A ) , F ) ∼ = Acc ω ( A , pt ( F )) . Being tensored over Acc ω means in a way to have a relative version of the Scottadjunction. Remark 3.21.
Among natural numbers, we find extremely familiar the followingformula, (30 × × × (5 × . Yet, this formula yields an important property of the category of sets. Indeed
Set istensored over itself and the tensorial structure is given by the product. The formulaabove tells us that the tensorial structure of
Set associates over its product.
Remark 3.22 (Associativity of ⊠ with respect to × ) . Recall that the category oftopoi has products, but they are very far from being computed as in Cat. Pitts hasshown [Pit85] that E × F ∼ = Cont( E ◦ , F ). This description, later rediscovered byLurie, is crucial to get a slick proof of the statement below. Proposition 3.23.
Let A be a finitely accessinble category. Then, A ⊠ ( E × F ) ≃ ( A ⊠ E ) × F . Proof.
We show it by direct computation. A ⊠ ( E × F ) ≃ Acc ω ( A , Cont( E ◦ , F )) ≃ Cat( A ω , Cont( E ◦ , F )) ≃ Cont( E ◦ , Cat( A ω , F )) ≃ Cont( E ◦ , Acc ω ( A , F )) ≃ ( A ⊠ F ) × E . (cid:3) ENERAL FACTS ON THE SCOTT ADJUNCTION 15
Remark 3.24.
Similarly to Rem. 3.21, the following display will appear completelytrivial, (30 × × × (1 × . Yet, we can get inspiration from it, to unveil an important simplification of thetensor A ⊠ E . We will show that it is enough to know the Scott topos S ( A ) tocompute A ⊠ E , at least when A is finitely accessible. Proposition 3.25 (Interaction between ⊠ and Scott) . Let A be a finitely access-inble category. Then, A ⊠ ( − ) ∼ = S ( A ) × ( − ) . Proof. A ⊠ ( − ) ∼ = A ⊠ ( Set ×− ) ∼ = ( A ⊠ Set ) × ( − ) . (cid:3) Proposition 3.26 (Powers and exponentiable Scott topoi) . Let A be a finitelyaccessible category. Then Topoi has powers with respect to A . Moreover, E A isgiven by the exponential topos E S ( A ) . Proof.
The universal property of the power object E A is expressed by the followingequation, Topoi( F , E A ) ∼ = Acc ω ( A , Topoi( F , E )) . Now, because we have tensors, this is saying that Topoi( F , E A ) ∼ = Topoi( A ⊠ F , E )) . Because of the previous proposition, we can gather this observation in the followingequation. Topoi( F , E A ) ∼ = Topoi( S ( A ) × F , E )) . This means that E A has the same universal property of the topos E S ( A ) and thusexists if and only if the latter exists. By the well known characterization of expo-nentiable topoi, this happens if and only if S ( A ) is continuous which is of coursetrue for presheaf categories. (cid:3) Toolbox
This section contains technical results on the Scott adjunction that will be ex-tensively employed for more qualitative results. We study the behavior of pt , S and η , trying to discern all their relevant properties. Before continuing, we briefly listthe main results that we will prove in order to facilitate the consultation.4.2 pt transforms geometric embeddings in fully faithful functors.4.4 pt transforms localic morphisms in faithful functors.4.6 S maps pseudo-epis (of Cat) to geometric surjections.4.10 S maps reflections to geometric embeddings.4.1.3 Introduces and studies the notion of topological embeddings between ac-cessible categories.4.15 η is faithful (and iso-full) if and only if A has a faithful (and iso-full) functorinto a finitely accessible category.4.1. Embeddings & surjections.Remark 4.1.
Observe that since S is a left adjoint, it preserve pseudo epimor-phisms, analogously pt preserves pseudo monomorphisms. Props. 4.2 and 4.6might be consequences of this observation, but we lack an explicit description ofpseudo monomorphisms and pseudo epimorphisms in both categories. Notice that,instead, 4.10 represents a non-trivial behavior of S . † On the behavior of pt . The functor pt behaves nicely with various notions of injective or locally injective geometric morphism.Geometric embeddings between topoi are a key object in topos theory. Intu-itively, they represent the proper notion of subtopos. It is a well known fact thatsubtopoi of a presheaf topos Set C correspond to Grothendieck topologies on C bijectively. Proposition 4.2. pt sends geometric embeddings to fully faithful functors. Proof.
This is a relatively trivial consequence of the fact that the direct imagefunctor is fully faithful but we shall include the proof in order to show a standardway of thinking. Let i ∗ : G ⇆ E : i ∗ be a geometric embedding. Recall, this meansprecisely that the E is reflective in G via this adjunction, i.e. the direct image isfully faithful. Let p, q : Set ⇒ E be two points, or equivalently let p ∗ , q ∗ : E ⇒ Set be two cocontinuous functors preserving finite limits. And let µ, ν : p ∗ ⇒ q ∗ be twonatural transformation between the points.The action of pt ( i ) on this data is the following. It maps p ∗ to p ∗ i ∗ while pt ( i )( µ )is defined by whiskering µ with i ∗ as pictured by the diagram below. p ∗ p ∗ i ∗ q ∗ q ∗ i ∗ νµ µ i ∗ ν i ∗ Now, observe that µ ∼ = µ i ∗ i ∗ because i ∗ i ∗ is isomorphic to the identity. Thisproves that pt ( i ) is faithful, in fact pt ( i )( µ ) = pt ( i )( ν ) means that µ i ∗ = ν i ∗ , thisimplies that µ i ∗ i ∗ = ν i ∗ i ∗ , and so µ = ν . A similar argument shows that pt ( i ) isfull (using that i ∗ is full). (cid:3) Localic topoi are those topoi that appear as the category of sheaves over a locale.Those topoi have a clear topological meaning and represent a quite concrete notionof generalized space. Localic morphisms are used to generalize the notion of localictopos; a localic morphism f : G → E attests that there exist an internal locale L in E such that G ≃ Sh ( L, E ). In accordance with this observation, a topos G is localicif and only if the essentially unique geometric morphism G → Set is localic.
Definition 4.3.
A morphism of topoi f : G → E is localic if every object in G isa subquotient of an object in the inverse image of f . Proposition 4.4. pt sends localic geometric morphisms to faithful functors. Proof.
Consider a localic geometric morphism f : G → E . We shall prove that pt f is faithful on points. In order to do so, let p, q : Set ⇒ E be two points, orequivalently let p ∗ , q ∗ : E ⇒ Set be two cocontinuous functors preserving finitelimits. And let µ, ν : p ∗ ⇒ q ∗ be two natural transformation between the points.We need to prove that if µ f ∗ = ν f ∗ , then µ = ν . In order to do so, let e bean object in E . Since f is a localic morphism, there is an object g ∈ G and anepimorphism l : f ∗ ( g ) ։ e . This is not quite true, we know that e is a subquotient of f ∗ ( g ), in the general case the proofgets a bit messier to follow, for this reason we will cover in detail just this case. ENERAL FACTS ON THE SCOTT ADJUNCTION 17 p ∗ ( e ) p ∗ i ∗ ( g ) q ∗ ( e ) q ∗ i ∗ ( g ) νµ µ i ∗ g ν i ∗ g p ∗ ( l ) q ∗ ( l ) Now, we know that µ ◦ p ∗ ( l ) = q ∗ ( l ) ◦ µ i ∗ g and ν ◦ p ∗ ( l ) = q ∗ ( l ) ◦ ν i ∗ g , becauseof the naturality of µ and ν . Since µ i ∗ = ν i ∗ , we get µ ◦ p ∗ ( l ) = q ∗ ( l ) ◦ µ i ∗ g = q ∗ ( l ) ◦ ν i ∗ g = ν ◦ p ∗ ( l ) . Finally observe that p ∗ ( l ) is an epi, because p ∗ preserves epis, and thus we cancancel it, obtaining the thesis. (cid:3) Proposition 4.5.
Let f : G → E be a geometric morphism. The following areequivalent. • For every point j : Set → E the pullback G × E Set has a point. • pt ( f ) is surjective on objects. Proof.
Trivial. (cid:3)
On the behavior of S . The functor S behaves nicely with respect to epis, asexpected. It does not behave nicely with any notion of monomorphism. In thenext section we study those accessible functors f such that S ( f ) is a geometricembedding. Proposition 4.6. S maps pseudo-epis (of Cat) to geometric surjections. Proof.
See [AEBSV01][4.2]. (cid:3)
Topological embeddings.
Definition 4.7.
Let f : A → B be a 1-cell in Acc ω . We say that f is a topologicalembedding if S ( f ) is a geometric embedding.This subsection is devoted to describe topological embeddings between accessiblecategories with directed colimits. The reader should expect this description to behighly nontrivial and rather technical, because S is a left adjoint and is not expectedto have nice behavior on any kind of monomorphism.Fortunately we will manage to provide some useful partial results. Let us listthe lemmas that we are going to prove.4.9 a necessary condition for a functor to admit a topological embedding intoa finitely accessible category4.10 a sufficient and quite easy to check criterion for a functor to be a topologicalembedding4.11 a full description of topological embeddings into finitely accessible cate-gories Remark 4.8.
Topological embeddings into finitely accessible categories i : A → Ind ( C ) are very important because S ( i ) will describe, by definition, a subtopos of Set C . This means that there exist a topology J on C such that S ( A ) is equivalentto Sh ( C, J ), this leads to concrete presentations of the Scott topos. † Lemma 4.9 (A necessary condition) . If A has a fully faithful topological embed-ding f : A → Ind ( C ) into a finitely accessible category, then η A : A → ptS ( A ) isfully faithful. Proof.
Assume that A has a topological embedding f : A → Ind ( C ) into a finitelyaccessible category. This means that S ( f ) is a geometric embedding. Now, we lookat the following diagram. A Ind ( C ) ptS ( A ) ptS ( Ind ( C )) fη A η Ind ( C ) ptS f η Ind ( C ) is an equivalence of categories, while 4.2 implies that ptS ( f )is fully faithful. Since also f is fully faithful, η A is forced to be fully faithful. (cid:3) Theorem 4.10.
Let i : A → B be a 1-cell in Acc ω exhibiting A as a reflectivesubcategory of B L : B ⇆ A : i. Then i is a topological embedding. Proof.
We want to show that S ( i ) is a geometric embedding. This is equivalentto show that the counit i ∗ i ∗ ( − ) ⇒ ( − ) is an isomorphism. Going back to 2.7, wewrite down the obvious computations, i ∗ i ∗ ( − ) ∼ = ( i ∗ ◦ r B ◦ ran i ◦ ι A )( − ) . Now, observe that since i has a left adjoint L the operator ran i just coincides with( − ) ◦ L , thus we can elaborate the previous equation as follows.( i ∗ ◦ r B ◦ ran i ◦ ι A )( − ) ∼ = ( i ∗ ◦ r B )( − ◦ L ) , Now, ( − ◦ L ) will preserve directed colimits because is the composition of a cocon-tinuous functor with a functor preserving directed colimits. This means that it isa fixed point of r B .( i ∗ ◦ r B )(( − ) ◦ L ) ∼ = i ∗ (( − ) ◦ L ) ∼ = ( − ) ◦ L ◦ i ∼ = ( − ) . The latter isomorphism is just the definition of reflective subcategory. This con-cludes the proof. (cid:3)
Theorem 4.11. f : A → Ind ( C ) is a topological embedding into a finitely accessiblecategory if and only if, for all p : A → Set preserving directed colimits, the followingequation holds (whenever well defined), lan i ( ran f ( p ) ◦ i ) ◦ f ∼ = p. Proof.
The result follows from the discussion below. (cid:3)
Remark 4.12 ( f ∗ and finitely accessible categories) . Given a 1-cell f : A → B inAcc ω , we experienced that it can be quite painful to give an explicit formula for thedirect image functor f ∗ . In this remark we improve the formula provided in 2.7 inthe special case that the codomain if finitely accessible. In order to do so we studythe diagram of 2.7. To settle the notation, call f : A → Ind ( C ) our object of studyand i : C → Ind ( C ) the obvious inclusion. ENERAL FACTS ON THE SCOTT ADJUNCTION 19 S A SInd ( C ) ¶ ( A ) ¶ ( Ind ( C )) ι A f ∗ ι Ind ( C ) f ∗ ran f lan f f ∗ We are use to this diagram from 2.7, where we learnt also the following formula f ∗ ∼ = r B ◦ ran f ◦ ι A . We now use the following diagram to give a better description of the previousequation. S A SInd ( C ) Set C ¶ ( A ) ¶ ( Ind ( C )) ι A f ∗ ι Ind ( C ) f ∗ i ∗ lan i lan i ran f lan f f ∗ i ∗ We claim that in the notations of the diagram above, we can describe the directimage f ∗ by the following formula, f ∗ ∼ = lan i ◦ i ∗ ◦ ran f ◦ ι A , this follows from the observation that r Ind ( C ) coincides with lan i ◦ i ∗ in the diagramabout.4.2. A study of the unit η A . This section is devoted to a focus on the unitof the Scott adjunction. We will show that good properties of η A are related tothe existence of finitely accessible representations of A . A weaker version of thefollowing proposition appeared in [Hen19][2.6]. Here we give a different proof, thatwe find more elegant and provide a stronger statement. Proposition 4.13.
The following are equivalent:(1) The unit at A of the Scott adjunction A → ptS A is faithful (and iso-full);(2) A admits a faithful (and iso-full) functor f : A → Ind ( C ) preserving directedcolimits; Proof. ⇒
2) Assume that η A is faithful. Recall that any topos admits a geometricembedding in a presheaf category, this is true in particular for S ( A ). Letus call ι some such geometric embedding ι : S ( A ) → Set X . Following 4.2and 2.13, pt ( ι ) is a fully faithful functor into a finitely accessible category pt ( ι ) : ptS ( A ) → Ind ( X ). Thus the composition pt ( ι ) ◦ η A is a faithfulfunctor into a finitely accessible category A → ptS A → Ind ( X ) . Obverse that if η A is iso-full, so is the composition pt ( ι ) ◦ η A .2) ⇒
1) Assume that A admits a faithful functor f : A → Ind ( C ) preserving directedcolimits. Now we apply the monad ptS obtaining the following diagram. † A Ind ( C ) ptS ( A ) ptS ( Ind ( C )) fη A η Ind ( C ) ptS f η Ind ( C ) is an equivalence of categories, thus ptS ( f ) ◦ η A is(essentially) a factorization of f . In particular, if f is faithful, so has tobe η A . Moreover, if f is iso-full and faithful, so must be η A , because thischaracterizes pseudo-monomorphisms in Cat (and by direct verification alsoin Acc ω ). (cid:3) Remark 4.14.
If we remove iso-fullness from the statement we can reduce therange of f from any finitely accessible category to the category of sets. Proposition 4.15.
The following are equivalent:(1) A admits a faithful functor f : A → Set preserving directed colimits.(2) A admits a faithful functor f : A → Ind ( C ) preserving directed colimits; Proof.
The proof is very simple. 1) ⇒
2) is completely evident. In order to prove2) ⇒ Ind ( C ) is finitely accessible, there is a faithful functor Y : Ind ( C ) → Set preserving directed colimits given by Y := a p ∈ C Ind ( C )( p, − ) . The composition g := Y ◦ f is the desired functor into Set . (cid:3) Acknowledgements
I am indebted to my advisor, Jiˇr´ı Rosick´y, for the freedom and the trust heblessed me with during these years, not to mention his sharp and remarkably bluntwisdom. I would like to thank Simon Henry, for his collaboration in those days inwhich this thesis was nothing but an informal conversation at the whiteboard. Iam grateful to Peter Arndt, for his sincere interest in my research, and the hint oflooking at the example of the geometric theory of fields.
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