aa r X i v : . [ m a t h . C T ] O c t TOWARDS HIGHER TOPOLOGY
IVAN DI LIBERTI † Abstract.
We categorify the adjunction between locales and topological spaces,this amounts to an adjunction between (generalized) bounded ionads andtopoi. We show that the adjunction is idempotent. We relate this adjunc-tion to the Scott adjunction, which was discussed from a more categoricalpoint of view in [Lib20b]. We hint that 0-dimensional adjunction inhabits thecategorified one.
Contents
1. Introduction 12. Spaces, locales and posets 43. Ionads, Topoi and Accessible categories 74. Sober ionads and topoi with enough points 125. From Isbell to Scott: categorified 126. Interaction 157. The κ -case 16Appendix A. Ionads 16Acknowledgements 21References 211. Introduction
Posets with directed suprema, topological spaces and locales provide three differentapproaches to geometry. Topological spaces are probably the most widespreadconcept and need no introduction, posets with directed suprema belong mostly todomain theory [AJ94], while locales are the main concept in formal topology andconstructive approaches to geometry [Vic07].LocTop Pos ω pt pt O SST
These three incarnations of geometry have been studied and related (as hinted bythe diagram above) in the literature, but they are far from representing the onlyexisting approaches to geometry. Grothendieck introduced topoi as a generalizationof locales of open sets of a topological space [BAG + † 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. † highest peaks in the 90’s. Since its introduction, topos theory has gained more andmore consensus and has shaped the language in which modern algebraic geometryhas been written. The notion of ionad, on the other hand, was introduced by Garnermuch more recently [Gar12]. Whereas a topos is a categorified locale, a ionad is acategorified topological space. Accessible categories were independently defined byLair and Rosick´y, even though their name was introduced by Makkai and Par´e in[MP89].The importance of accessible (and locally presentable) categories steadily growedand today they form an established framework to do category theory in the dailypractice of the working mathematician. While ionads and topoi are arguably anapproach to higher topology (whatever higher topology is), the significance of acces-sible categories with directed colimits is very far from being topological , no matterwhich notion of the word we choose. In this paper we study the relationship be-tween these two different approaches to higher topology and use this geometricintuition to analyze accessible categories with directed colimits.We build on the analogy (and the existing literature on the topic) between plain and higher topology and offer a generalization of the diagram above to accessiblecategories with directed colimits, ionads and topoi .TopoiBIon Acc ω pt pt O SST
The functors that regulate the interaction between these different approaches togeometry are the main characters of the preprint. These are the
Scott adjunc-tion , which relates accessible categories with directed colimits to topoi and the categorified Isbell adjunction , which relates bounded ionads to topoi. Since wehave a quite good understanding of the topological case, we use this intuition toguess and infer the behavior of its categorification. In the diagram above, the onlyfunctor that has already appeared in the literature is O , in [Gar12]. The adjunction S ⊣ pt was achieved in collaboration with Simon Henry and has appeared in [Hen19]and [Lib20b]. Idempotency.
The adjunction between topological spaces and locales is idempo-tent, and it restricts to the well known equivalence of categories between localeswith enough points and sober spaces. We obtain an analogous result for the ad-junction between bounded ionads and topoi, inducing an equivalence between sober(bounded) ionads and topoi with enough points. Studying the Scott adjunction ismore complicated, and in general it does not behave as well as the categorifiedIsbell duality does, yet we manage to describe the class of those topoi which arefixed by the Scott-comonad.
Relevance and Impact: Stone-like dualities & higher semantics.
Thistopological picture fits the pattern of Stone-like dualities. As the latter is re-lated to completeness results for propositional logic, its categorification is relatedto syntax-semantics dualities between categories of models and theories. Indeeda logical intuition on topoi and accessible categories has been available for manyyears [LM94]. A topos can be seen as a placeholder for a geometric theory, while anaccessible category can be seen as a category of models of some infinitary theory.In a further work [Lib20a] we introduce a new point of view on ionads, defining ionads of models of a geometric sketch. This approach allows us to entangle the
OWARDS HIGHER TOPOLOGY 3 theory of classifying topoi with the Scott and Isbell adjunctions providing severalcomparison results in this direction.
Structure.
The paper is divided in seven sections and an appendix,Sec. 2 we recall the concrete topology on which the analogy is built: the Isbellduality, relating topological spaces to locales. We also relate the Isbellduality to Scott’s seminal work on the Scott topology, this first part iscompletely motivational and expository. This section contains the posetalversion of sections 3, 4 and 5.Sec. 3 We introduce the higher dimensional analogs of topological spaces, localesand posets with directed colimits: ionads, topoi and accessible categorieswith directed colimits. We categorify the Isbell duality building on Garner’swork on ionads, and we relate the Scott adjunction to its posetal analog.
Theorem (Thm. 3.2.11, Categorified Isbell adjunction) . There isa 2-adjunction, O : BIon ⇆ Topoi : pt . The left adjoint of this adjunction was found by Garner; in order to finda right adjoint, we must allow for large ionads (
Con. 3.2.8 ).Sec. 4 We study the the categorified version of Isbell duality. This amounts tothe notion of sober ionad and topos with enough points. Building on the idempotency of the categorified Isbell adjunction (Thm. 4.0.3) .Sec. 5 We use the previous section to derive properties of the Scott adjunction.We describe those topoi for which the counit of the Scott adjunction is anequivalence of categories (
Thm. 5.2.3 ).Sec. 6 We show that the analogy on which the chapter is built is deeply moti-vated and we show how to recover the content of the first section from thefollowing ones (
Thm. 6.0.2 ).Sec. 7 In the last section we provide an expected generalization of the second oneto κ -topoi and κ -ionads.Ap. A We recall Garner’s definition of ionad and we generalize it to large (butlocally small) finitely pre-cocomplete categories. We provide some technicaltools.1.1. 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 we wouldlike to settle some notation.(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) A Scott topos is a topos of the form S ( A ).(10) ¶ ( X ) is the category of small copresheaves of X . Notation 1.1.1 (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 category IVAN DI LIBERTI † of sheaves over τ ; in this sense f presents the topos as the category of sheaves overthe site ( C, τ f ). 2. Spaces, locales and posets
Our topological safari will start from the celebrated adjunction between locales andtopological spaces. This was first observed by Isbell, whence the name Isbell ad-junction/duality. Unfortunately this name is already taken by the dual adjunctionbetween presheaves and co-presheaves; this sometimes leads to some terminologicalconfusion. The two adjunctions are similar in spirit, but do not generalize, at leastnot apparently, in any common framework. This first subsection is mainly expos-itory and we encourage the interested reader to consult [Joh86] and [LM94][Chap.IX] for an extensive introduction. The aim of the subsection is not to introduce thereader to these results and objects, it is instead to organize them in a way that isuseful to our purposes. More precise references will be given within the section.2.1.
Spaces, Locales and Posets.
This subsection tells the story of the diagrambelow. Let us bring every character to the stage.LocTop Pos ω O SST
Remark 2.1.1 (The categories) . Loc is the category of locales. It is defined to be the opposite category of frames,where objects are frames and morphisms are morphisms of frames.Top is the category of topological spaces and continuous mappings betweenthem.Pos ω is the category of posets with directed suprema and functions preservingdirected suprema. Remark 2.1.2 (The functors) . The functors in the diagram above are well knownto the general topologist; we proceed to a short description of them. O associates to a topological space its frame of open sets and to each contin-uous function its inverse image. ST equips a poset with directed suprema with the Scott topology [Joh86][Chap.II, 1.9]. This functor is fully faithful, i.e. a function is continuous withrespect to the Scott topologies if and only if preserves suprema. S is the composition of the previous two; in particular the diagram abovecommutes. Remark 2.1.3 (The Isbell duality and a posetal version of the Scott adjunction) . Both the functors O and S have a right adjoint; we indicate them by pt and pt ,which in both cases stands for points . LocTop Pos ω pt pt O SST
OWARDS HIGHER TOPOLOGY 5
In the forthcoming remarks we will motivate the reason for this clash of names;indeed the two functors look alike. The adjunction on the left is
Isbell duality ,while the one on the right was not named yet to our knowledge and we will referto it as the (posetal) Scott adjunction . Let us mention that there exists a naturaltransformation ι : ST ◦ pt ⇒ pt , which will be completely evident from the description of the functors. We will saymore about ι ; for the moment we just want to clarify that there is no reason tobelieve (and indeed it would be a false belief) that ι is an isomorphism. Remark 2.1.4 (The Isbell duality) . An account of the adjunction O ⊣ pt can befound in the very classical reference [LM94][IX.1-3]. While the description of O isrelatively simple, (it associates to a topological space X its frame of opens O ( X )), pt is more complicated to define. It associates to a locale L its sets of formal points Loc( T , L ) equipped with the topology whose open sets are defined as follows: forevery l in L we pose, V ( l ) := { p ∈ Frm( L, T ) : p ( l ) = 1 } . Further details may be found in [LM94][IX.2]. In the literature, such an adjunctionthat is induced by a contravariant hom-functor is called schizophrenic. The specialobject in this case is 2 that inhabits both the categories under the alias of Sierpi´nskispace in the case of Top and the terminal object in the case of Frm. A detailedaccount on this topic can be found in [LM94][IX.3, last paragraph].
Remark 2.1.5 (The (posetal) Scott adjunction) . To our knowledge, the adjunction S ⊣ pt does not appear explicitly in the literature. Let us give a description of boththe functors involved. pt The underlying set of pt ( L ) is the same as for Isbell duality. Its posetalstructure is inherited from T ; in fact Frm( L, T ) has a natural poset structurewith directed unions given by pointwise evaluation [Vic07][1.11]. S Given a poset P , its Scott locale S ( P ) is defined by the frame Pos ω ( P, T ).It’s quite easy to show that this poset is a locale.Observe that also this adjunction is a dual one, and is induced by precisely thesame object as for Isbell duality. Remark 2.1.6 (An unfortunate naming) . There are several reasons for which weare not satisfied with the naming choices in this paper. Already in the topologicalcase, Isbell duality (or adjunction) is a very much overloaded name, and laterin the text we will categorify this adjunction, calling it categorified Isbell duality ,which propagates an unfortunate choice. Also the name of Scott for the Scottadjunction is not completely proper, because he introduced the Scott topology on aset, providing the functor ST . In the previous chapter, we called Scott adjunctionthe categorification of the posetal Scott adjunction in this section, propagating thisincorrect attribution. Yet, we did not manage to find better options, and thus wewill stick to these choices.2.2. Sober spaces and spatial locales.
The Isbell adjunction is a fascinatingconstruction that in principle could be an equivalence of categories.
Shouldn’t aspace be precisely the space of formal points of its locale of open sets?
It turnsout that the answer is in general negative; this subsection describes how far thisadjunction is from being an equivalence of categories. T is the boolean algebra { < } . IVAN DI LIBERTI † Remark 2.2.1 (Unit and counit) . Given a space X the unit of the Isbell adjunction η X : X → ( pt ◦ O )( X ) might not be injective. This is due to the fact that if twopoints x, y in X are such that cl ( x ) = cl ( y ), then η X will confuse them. Remark 2.2.2 (Sober spaces and spatial locales) . In the classical literature aboutthis adjunction people have introduced the notion of sober space and locale withenough points, these are precisely those objects on which the (co)unit is an iso.It turns out that even if η X is not always an iso η pt ( L ) is always an iso and thischaracterizes those η that are isomorphisms. An analogue result is true for thecounit. Remark 2.2.3 (Idempotency) . The technical content of the previous remark issummarized in the fact that the Isbell adjunction O ⊣ pt is idempotent; this isproved in [LM94][IX.3][Prop. 2, Prop. 3 and Cor. 4.]. It might look like this resulthas no qualitative meaning. Instead it means that given a locale L , the locale ofopens sets of its points O pt ( L ) is the best approximation of L among spatial locales,namely those that are the locale of opens of a space. The same observation is truefor a space X and the formal points of its locale of open sets pt O ( X ). In the nexttwo proposition we give a more categorical and more concrete incarnation of thisremark. Theorem 2.2.4 ([LM94][IX.3.3]) . The following are equivalent:(1) L has enough points;(2) the counit ǫ L : ( O ◦ pt )( L ) → L is an isomorphism of locales;(3) L is the locale of open sets O ( X ) of some topological space X . Theorem 2.2.5.
The subcategory of locales with enough points is coreflective inthe category of locales.
Remark 2.2.6.
Similarly, the subcategory of sober spaces is reflective in the cate-gory of spaces. This is not surprising, it’s far form being true that any adjunction isidempotent, but it is easy to check that given an adjunction whose induced comonadis idempotent, so must be the induced monad (and vice-versa).2.3.
From Isbell to Scott: topology.Remark 2.3.1 (Relating the Scott construction to the Isbell duality) . Going backto the (non-)commutativity of the diagram in Rem. 2.1.3, we observe that thereexists a natural transformation ι : ST ◦ pt ⇒ pt .LocTop Pos ω pt pt ι ST The natural transformation is pointwise given by the identity (for the underlyingset is indeed the same), and witnessed by the fact that every Isbell-open (Rem.2.1.4) is a Scott-open (Rem. 2.1.5). This observation is implicit in [Joh86][II, 1.8].
Remark 2.3.2 (Scott is not always sober) . In principle ι might be an isomorphism.Unfortunately it was shown by Johnstone in [Joh81] that some Scott-spaces are notsober. Since every space in the image of pt is sober, ι cannot be an isomorphismat least in those cases. Yet, Johnstone says in [Joh81] that he does not know anyexample of a complete lattice whose Scott topology is not sober. Thus it is naturalto conjecture that when pt ( L ) is complete, then ι L is an isomorphism. We will notonly show that this is true, but even provide a generalization of this result later. OWARDS HIGHER TOPOLOGY 7
Remark 2.3.3 (Scott from Isbell) . Let us conclude with a version of [LM94][IX.3.3]for the Scott adjunction. This has guided us in understanding the correct idempo-tency of the Scott adjunction.
Theorem 2.3.4 (Consequence of [LM94][IX.3.3]) . The following are equivalent:(1) L has enough points and ι is an isomorphism at L ;(2) the counit of the Scott adjunction is an isomorphism of locales at L . Proof.
It follows directly from [LM94][IX.3.3] and the fact that ST is fully faithful. (cid:3) Ionads, Topoi and Accessible categories
Now we come to the 2-dimensional counterpart of the previous section. As in theprevious one, this section is dedicated to describing the properties of a diagram.TopoiBIon Acc ω O SST
Motivations.
There are no doubts that we drew a triangle which is quitesimilar to the one in the previous section, but in what sense are these two trianglesrelated?
There is a long tradition behind this question and too many papers shouldbe cited. In this very short subsection we provide an intuitive account regardingthis question.
Remark 3.1.1 (Replacing posets with categories) . There is a well known analogy between the category of posets Pos and the category of categories
Cat . A part of thisanalogy is very natural: joins and colimits, meets and limits, monotone functionsand functors. Another part might appear a bit counter intuitive at first sight. Theposet of truth values T = { < } plays the rˆole of the category of sets. Theinclusion of a poset i : P → T P ◦ in its poset of ideals plays the same rˆole of theYoneda embedding. Pos Cat P → T P ◦ C → Set C ◦ joins colimitsmeets limitsmonotone function functor Remark 3.1.2 (Pos ω and Acc ω ) . Following the previous remark one would betempted to say that posets with directed colimits correspond to categories withdirected colimits. Thus, why put the accessibility condition on categories?
Thereason is that in the case of posets the accessibility condition is even stronger, evenif hidden. In fact a poset is a small (!) poclass (partially ordered set), and poclasseswith directed joins would be the correct analog of categories with directed colim-its. Being accessible is a way to have control on the category without requestingsmallness, which would be too strong an assumption. It is much more than an analogy, but this remark is designed to be short, motivational andinspirational. Being more precise and mentioning enriched categories over truth values would notgive a more accessible description to the generic reader.
IVAN DI LIBERTI † Remark 3.1.3 (Locales and Topoi) . Quite surprisingly the infinitary distributivityrules which characterizes locales has a description in term of the posetal
Yoneda inclusion. Locales can be described as those posets whose (Yoneda) inclusion hasa left adjoint preserving finite joins, L : T P ◦ ⇆ P : i. In the same fashion, Street [Str81] proved that Topoi can be described as thosecategories (with a generator) whose Yoneda embedding G → Set G ◦ has a left adjointpreserving finite limits. In analogy with the case of locales this property is reflectedby a kind of interaction been limits and colimits which is called descent . In thissense a topos is a kind of Set -locale, while a locale is a T -locales. In the nextsection we will show that there is an interplay between these two notions of locale . Remark 3.1.4 (Spaces and Ionads) . While topoi are the categorification of locales,ionads are the categorification of topological spaces. Recall that a topological space,after all, is just a set X together with the data of its interior operator int : 2 X → X . This is an idempotent operator preserving finite meets. We will see that ionads aredefined in a very similar way, following the pattern of the previous remarks.3.2.
Categorification.
Now that we have given some motivation for this to bethe correct categorification of the Isbell duality, we have to present the involvedingredients in more mathematical detail. A part of the triangle in this section isjust the Scott adjunction, that we understood quite well in the previous section.Here we have to introduce ionads and all the functors involving them.TopoiBIon Acc ω O SST
Remark 3.2.1 (The ( -)categories) . Topoi is the 2-category of topoi and geometric morphisms.BIon is the 2-category of (generalized) bounded ionads.Acc ω is the 2-category of accessible categories with directed colimits and functorspreserving them.3.2.1. The functors.
Remark 3.2.2 ( O ) . Let us briefly recall the relevant definitions of App. A. Ageneralized bounded ionad X = ( X, Int) is a (possibly large, locally small andfinitely pre-cocomplete) category X together with a comonad Int : ¶ ( X ) → ¶ ( X )preserving finite limits whose category of coalgebras is a Grothendieck topos. O was described by Garner in [Gar12][Rem. 5.2], it maps a bounded ionad to itscategory of opens, that is the category of coalgebras for the interior operator. Construction 3.2.3 ( S ) . We recall the construction of S from [Hen19] and [Lib20b].Let A be an accessible category with directed colimits. S ( A ) is defined as the cat-egory the category of functors preserving directed colimits into sets. S ( A ) = Acc ω ( A , Set ) . Let f : A → B be a 1-cell in Acc ω , the geometric morphism S f is defined byprecomposition as described below. OWARDS HIGHER TOPOLOGY 9 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 . Remark 3.2.4 ( S ( A ) is coreflective in ¶ ( A ) ) . The inclusion i A : S ( A ) → ¶ ( A ) has a right adjoint r A ; unfortunately this result is true for a rather technical argu-ment. By a general result of Kelly, i A has a right adjoint if and only if lan i A (1 S ( A ) )exists and i A preserves it. Since S ( A ) is small cocomplete, if lan i A (1 S ( A ) ) exists, itmust be pointwise and thus i will preserve it because it is a cocontinuous functor.Thus it is enough to prove that lan i A (1 S ( A ) ) exists. Anyone would be tempted toapply [Bor94a][3.7.2], unfortunately S ( A ) is not a small category. In order to cutdown this size issue, we use the fact that S ( A ) is a topos and thus have a densegenerator 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 [Bor94a][3.7.2], because G isa small category. Construction 3.2.5 ( ST ) . The construction is based on Con. 3.2.3 and Rem 3.2.4,we map A to the bounded ionad ( A , r A i A ), as described in Rem. 3.2.4. A functor f : A → B is sent to the morphism of ionads ( f, f ♯ ) below, where f ♯ coincides withthe inverse image of S ( f ). S ( B ) S ( A ) ¶ ( B ) ¶ ( A ) f ♯ f ∗ Coherently with the previous section, it is quite easy to notice that S ∼ = O ◦ ST .Let us show it, O ◦ ST ( A ) = O ( ST ( A )) 3 . . coAlg ( r A i A ) ∼ = S ( A ) . Points.
Remark 3.2.6 (Categorified Isbell duality and the Scott Adjunction) . As in theprevious section, both the functors O and S have a right (bi)adjoint. We indicatethem by pt and pt , which in both cases stands for points. pt has of course beenintroduced in the previous chapter and correspond to the right adjoint in the Scottadjunction. The other one will be more delicate to describe. ¶ ( A ) is the category of small copresheaves over A . We discuss its most relevat properties inApp. A. † TopoiBIon Acc ω pt pt O SST
This sub-subsection will be mostly dedicated to the construction of pt and to showthat it is a right (bi)adjoint for O . Let us mention though that there exists a naturalfunctor ι : ST ◦ pt ⇒ pt which is not in general an equivalence of categories. Remark 3.2.7 (The -functor pt ) . The functor of points pt belongs to the litera-ture since quite some time, pt is the covariant hom functor Topoi( Set , − ). It mapsa 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 [Bor94b][Cor. 4.3.2], while the fact that pt ( f ) preserves directed colimitsfollows trivially from the definition. Construction 3.2.8 (Topoi
Bion: every topos induces a generalized boundedionad over its points) . For a topos E , there exists a natural evaluation pairing ev : E × pt ( E ) → Set , mapping the couple ( e, p ) to its evaluation p ∗ ( e ). This construction preserves col-imits and finite limits in the first coordinate, because p ∗ is an inverse image functor.This means that its mate functor ev ∗ : E → Set pt ( E ) , preserves colimits and finitelimits. Moreover ev ∗ ( e ) preserves directed colimits for every e ∈ E . Indeed, ev ∗ ( e )(colim p ∗ i ) ∼ = (colim p ∗ i )( e ) ( ⋆ ) ∼ = colim(( p ∗ i )( e )) ∼ = colim( ev ∗ ( e )( p ∗ i )) . where ( ⋆ ) is true because directed colimits of points are computed pointwise. Thus,since the category of points of a topos is always accessible (say λ -accessible) and ev ∗ ( e ) preserves directed colimits, the value of ev ∗ ( e ) is uniquely individuated byits restriction to pt ( E ) λ . Thus, ev ∗ takes values in ¶ ( pt ( E )). Since a topos is atotal category, ev ∗ must have a right adjoint ev ∗ , and we get an adjunction, ev ∗ : E ⇆ ¶ ( pt ( E )) : ev ∗ . Since the left adjoint preserves finite limits, the comonad ev ∗ ev ∗ is lex and thusinduces a ionad over pt ( E ). This ionad is bounded, this follows from a carefulanalysis of the discussion above. Indeed Set pt ( E ) λ is lex-coreflective in ¶ ( pt ( E ))and S → E → Set pt ( E ) λ ⇆ ¶ ( pt ( E )), where S is a site of presentation of E , satisfiesthe hypotheses of Prop. A.3.18(3). Construction 3.2.9 (Topoi
Bion: every geometric morphism induces a mor-phism of ionads) . Observe that given a geometric morphism f : E → F , pt ( f ) : pt ( E ) → pt ( F ) induces a morphism of ionads ( pt ( f ) , pt ( f ) ♯ ) between pt ( E ) and pt ( F ). In order to describe pt ( f ) ♯ , we invoke Prop. A.3.21[(a)]. Thus, it is enough For a total category the adjoint functor theorem reduces to check that the candidate leftadjoint preserves colimits.
OWARDS HIGHER TOPOLOGY 11 to provide a functor making the diagram below commutative (up to natural iso-morphism).
F E ¶ ( pt ( F )) ¶ ( pt ( E )) ev ∗ F f ⋄ ev ∗ E pt ( f ) ∗ Indeed such a functor exists and coincides with the inverse image f ∗ of the geometricmorphism f . Remark 3.2.10 (The -functor pt ) . pt ( E ) is defined to be the ionad ( pt ( E ) , ev ∗ ev ∗ ),as described in the two previous remarks. Theorem 3.2.11 (Categorified Isbell adjunction, O ⊣ pt ) . O : BIon ⇆ Topoi : pt Proof.
We provide the unit and the counit of this adjunction. This means that weneed to provide geometric morphisms ρ : O pt ( E ) → E and morphisms of ionads λ : X → pt O X . Let’s study the two problems separately.( ρ ) As in the case of any geometric morphism, it is enough to provide theinverse image functor ρ ∗ : E → O pt ( E ). Now, recall that the interioroperator of pt ( E ) is induced by the adjunction ev ∗ : E ⇆ ¶ ( pt ( E )) : ev ∗ asdescribed in the remark above. By the universal property of the categoryof coalgebras, the adjunction U : O pt ( E ) ⇆ ¶ ( pt ( E )) : F is terminal amongthose adjunctions that induce the comonad ev ∗ ev ∗ . This means that thereexists a functor ρ ∗ lifting e ∗ along U as in the diagram below. E O pt ( E ) ¶ ( pt ( E )) ev ∗ ρ ∗ U It is easy to show that ρ ∗ is cocontinuous and preserves finite limits and isthus the inverse image functor of a geometric morphism ρ : O pt ( E ) → E as desired.( λ ) Recall that a morphism of ionads λ : X → pt O X is the data of a functor λ : X → pt O X together with a lifting λ ♯ : O X → O pt O X . We only provide λ : X → pt O X , λ ♯ is induced by Prop. A.3.21. Indeed such a functor isthe same of a functor λ : X → Cocontlex( O X , Set ). Define, λ ( x )( s ) = ( U ( s ))( x ) . From a more conceptual point of view, λ is just given by the compositionof the functors, X eval −→ Cocontlex( ¶ ( X ) , Set ) −◦ U −→ Cocontlex( O X , Set ) . (cid:3) Theorem 3.2.12 ([Hen19][Prop. 2.3], [Lib20b][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 . † Sober ionads and topoi with enough points
In this section we show that the categorified Isbell adjunction is idempotent, pro-viding a categorification of Subsec. 2.2. The notion of sober ionad is a bit unsat-isfactory and lacks an intrinsic description. Topoi with enough points have beenstudied very much in the literature. Let us give (or recall) the two definitions.
Definition 4.0.1 (Sober ionad) . A ionad is sober if λ is an equivalence of ionads. Definition 4.0.2 (Topos with enough points) . A topos has enough points if theinverse image functors from all of its points are jointly conservative.
Theorem 4.0.3 (Idempotency of the categorified Isbell duality) . The followingare equivalent:(1) E has enough points;(2) ρ : O pt ( E ) → E is an equivalence of categories;(3) E is equivalent to a topos of the form O ( X ) for some bounded ionad X . Proof. (1) ⇒ (2) Going back to the definition of ρ in Thm. 3.2.11, it’s enough to show that ev ∗ is comonadic. Since it preserves finite limits, it’s enough to show thatit is conservative to apply Beck’s (co)monadicity theorem. Yet, that is justa reformulation of having enough points .(2) ⇒ (3) Trivial.(3) ⇒ (1) [Gar12][Rem. 2.5]. (cid:3) Theorem 4.0.4.
The following are equivalent:(1) X is sober;(2) X is of the form pt ( E ) for some topos E . Proof.
For any adjunction, it is enough to show that either the monad or thecomonad is idempotent, to obtain the same result for the other one. (cid:3) From Isbell to Scott: categorified
This section is a categorification of its analog Subsec. 2.3 and shows how to inferresults about the tightness of the Scott adjunction from the Isbell adjunction. Wementioned in Rem. 3.2.6 that there exists a natural transformation as described bythe diagram below. TopoiBIon Acc ω pt pt ι ST Let us describe it. Spelling out the content of the diagram, ι should be a morphismof ionads ι : STpt ( E ) → pt ( E ) . Recall that the underling category of these two ionads is pt ( E ) in both cases. Wedefine ι to be the identity on the underlying categories, ι = (1 pt ( E ) , ι ♯ ) while ι ♯ isinduced by the following assignment defined on the basis of the ionad E → Spt ( E ), ι ♯ ( x )( p ) = p ∗ ( x ) . OWARDS HIGHER TOPOLOGY 13
Remark 5.0.1 ( ι ♯ is the counit of the Scott adjunction) . The reader might havenoticed that ι ♯ is precisely the counit of the Scott adjunction. Theorem 5.0.2 (From Isbell to Scott, cheap version) . The following are equivalent:(1) E has enough points and ι is an equivalence of ionads.(2) The counit of the Scott adjunction is an equivalence of categories. Proof.
This is completely obvious from the previous discussion. (cid:3)
This result is quite disappointing in practice and we cannot accept it as it is. Yet,having understood that the Scott adjunction is not the same as Isbell one was veryimportant conceptual progress in order to guess the correct statement for the Scottadjunction. In the next section we provide a more useful version of the previoustheorem.5.1.
Covers.
In order to provide a satisfying version of Thm. 5.0.2, we need to in-troduce a tiny bit of technology, namely finitely accessible covers. For an accessiblecategory with directed colimits A a (finitely accessible) cover L : Ind ( C ) → A will be one of a class of cocontinuous (pseudo)epimorphisms (in Cat) having manygood properties. They will be helpful for us in the discussion. Covers were intro-duced for the first time in [BR12][4.5] and later used in [LR14][2.5]. Remark 5.1.1 (Generating covers) . Every object A in Acc ω has a (proper classof) finitely accessible cover. Let A be κ -accessible. Let us focus on the followingdiagram. A κ Ind A κ A α ι lan α ι A ( ι − , − ) A ( ι − , − ), also known as the nerve of f , is fully faithful because A κ is a (dense)generator in A . This functor does not lie in Acc ω because it is just κ -accessible ingeneral. lan α ( ι ) exists by the universal property of of the Ind completion, indeed A has directed colimits by definition. For a concrete perspective lan α ( ι ) is evaluatinga formal directed colimit on the actual directed colimit in A . These two maps yieldan adjunction lan α ( ι ) ⊣ A ( ι − , − )that establishes A as a reflective embedded subcategory of Ind A κ . Since lan α ( ι ) isa left adjoint, it lies in Acc ω . Definition 5.1.2.
When A is a κ -accessible category we will indicate with L κ A themap that here we indicated with lan α ( ι ) in the previous remark and we call it a cover of A . L κ A : Ind A κ → A . Notation 5.1.3.
When it’s evident from the context, or it is not relevant at all wewill not specify the cardinal on the top, thus we will just write L A instead of L κ A . Remark 5.1.4.
When λ ≥ κ , L λ A ∼ = L κ A L λκ for some transition map L λκ . We didnot find any application for this, thus we decided not to go in the details of thisconstruction. Remark 5.1.5.
This construction appeared for the first time in [BR12][4.5] andlater in [LR14][2.5], where it is presented as the analogue of Shelah’s presentationtheorem for AECs [LR14][2.6]. The reader that is not familiar with formal category † theory might find the original presentation more down to earth. In [LR14][2.5] itis also shown that under interesting circumstances the cover is faithful.5.2. On the (non) idempotency of the Scott adjunction.
This subsectionprovides a better version of Thm. 5.0.2. It is based on a technical notion (topolog-ical embedding) that we have defined and studied in the Toolbox of [Lib20b]. Letus recall the definition, we refer to that paper for the relevant results.
Definition 5.2.1.
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. Theorem 5.2.2.
A Scott topos G ∼ = S ( A ) has enough points. Proof.
It is enough to show that G admits a geometric surjection from a presheaftopos [Joh02][2.2.12]. Let κ be a cardinal such that A is κ -accessible. We claimthat the 1-cell S ( L κ A ) described in Rem. 5.1.2 is the desired geometric surjectionfor the Scott topos S ( A ). By [Lib20b][Rem. 2.13], the domain of S ( L κ A ) is indeed apresheaf topos. By [Lib20b][Prop. 4.6], it is enough to prove that L κ A is a pseudoepimorphism in Cat. But that is obvious, because it has even a section in Cat,namely A ( ι, (cid:3) Theorem 5.2.3.
The following are equivalent.(1) The counit ǫ : Spt ( E ) → E is an equivalence of categories.(2) E has enough points and for all presentations i ∗ : Set X ⇆ E : i ∗ , pt ( i ) is atopological embedding.(3) E has enough points and there exists a presentation i ∗ : Set X ⇆ E : i ∗ such that pt ( i ) is a topological embedding. Proof.
As expected, we follow the proof strategy hinted at by the enumeration.1) ⇒
2) By Thm. 5.2.2, E has enough points. We only need to show that for allpresentations i ∗ : Set X ⇆ E : i ∗ , pt ( i ) is a topological embedding. In orderto do so, consider the following diagram, Spt E Spt
Set X E Set Xǫ E Spt ( i ) ǫ Set X i By [Lib20b][Rem. 2.13] and the hypotheses of the theorem, one obtains that
Spt ( i ) is naturally isomorphic to a composition of geometric embedding Spt ( i ) ∼ = ǫ − Set X ◦ i ◦ ǫ E , and thus is a geometric embedding. This shows precisely that pt ( i ) is atopological embedding.2) ⇒
3) Obvious.3) ⇒
1) It is enough to prove that ǫ E is both a surjection and a geometric embeddingof topoi. ǫ E is a surjection, indeed since E has enough points, there exista surjection q : Set X ։ E , now we apply the comonad Spt and we look atthe following diagram,
Spt
Set X Set X Spt
E E
Spt ( q ) ǫ Set X qǫ E OWARDS HIGHER TOPOLOGY 15
Now, the counit arrow on the top is an isomorphism, because
Set X is apresheaf topos. Thus ǫ E ◦ ( Spt )( q ) is (essentially) a factorization of q . Since q is a geometric surjection so must be ǫ E . In order to show that ǫ E is ageometric embedding, we use again the following diagram over the givenpresentation i . Spt E Spt
Set X E Set Xǫ E Spt ( i ) ǫ Set X i This time we know that
Spt ( i ) and i are geometric embeddings, and thus ǫ E has to be so. (cid:3) Remark 5.2.4.
The version above might look like a technical but not very use-ful improvement of Thm. 5.0.2. Instead, in the following Corollary we prove anon-trivial result based on a characterization (partial but useful) of topologicalembeddings contained in the Toolbox.
Corollary 5.2.5.
Let E be a topos with enough points, together with a presenta-tion i : E → Set C . If pt ( E ) is complete and pt ( i ) preserves limits. Then ǫ E is anequivalence of categories. Proof.
We verify the condition (3) of the previous theorem. Since pt ( i ) preserveall limits, fully faithful, and pt ( E ) must be complete, pt ( i ) has a left adjoint bythe adjoint functor theorem. This establishes pt E as a reflective subcategory of pt Set C . By [Lib20b][Thm. 4.10], pt ( i ) must be a topological embedding. (cid:3) Remark 5.2.6 (Scott is not aways sober) . In [Joh81], Johnstone’s observes thatwhen a poset is (co)complete its Scott topology is generically sober. The previouscorollary is very much in this spirit and shows that the the category of points ofthe topos is complete, it is very likely that Scott is sober.6.
Interaction
In this section we shall convince the reader that the posetal version of the Scott-Isbell story embeds in the categorical one.Loc TopoiTop BIonPos ω Acc ω Sh ι We have no applications for this observation, thus we do not provide all the detailsthat would amount to an enormous amount of functors relating all the categoriesthat we have mentioned. Yet, we show the easiest aspect of this phenomenon. Letus introduce and describe the following diagram,Loc TopoiPos ω Acc ω Shpt pt i6 IVAN DI LIBERTI † Remark 6.0.1 ( Sh and i ) . Sh It is well known that the sheafification functor Sh : Loc → Topoi establishesLoc as a full subcategory of Topoi in a sense made precise in [LM94][IX.5,Prop. 2 and 3]. i This is very easy to describe. Indeed any poset with directed supremais an accessible category with directed colimits and a function preservingdirected suprema is precisely a functor preserving directed colimits.
Proposition 6.0.2.
The diagram above commutes.
Proof.
This is more or less tautological from the point of view of [LM94][IX.5, Prop.2 and 3]. In fact, pt ( L ) = Loc( T , L ) ∼ = Topoi( Sh ( T ) , Sh ( L )) ∼ = pt ( Sh ( L )) . (cid:3) The κ -case Unsurprisingly, it is possible to generalize the whole content of this chapter to the κ -case. The notion of κ -ionad is completely straightforward and every constructionlifts to infinite cardinals without any effort. For the sake of completeness, we reportthe κ -version of the main theorems that we saw in the chapter, but we omit theproof, which would be identical to the finitary case. Remark 7.0.1.
The following diagram describes the κ -version of the relevant ad-junctions. κ -Topoi κ -BIon Acc κ pt κ pt κ O κ S κ ST κ Theorem 7.0.2 (Idempotency of the categorified κ -Isbell duality) . Let E be a κ -topos. The following are equivalent:(1) E has enough κ -points;(2) ρ : O κ pt κ ( E ) → E is an equivalence of categories;(3) E is of the form O κ ( X ) for some bounded κ -ionad X . Theorem 7.0.3.
Let E be a κ -topos. The following are equivalent.(1) The counit ǫ : S κ pt κ ( E ) → E is an equivalence of categories.(2) E has enough κ -points and for all presentations i ∗ : Set X ⇆ E : i ∗ , pt κ ( i )is a topological embedding.(3) E has enough κ -points and there exists a presentation i ∗ : Set X ⇆ E : i ∗ such that pt ( i ) is a topological embedding. Appendix A. Ionads
A.1.
Garner’s definitions.Definition A.1.1 (Ionad) . An ionad X = ( X, Int) is a set X together with acomonad Int : Set X → Set X preserving finite limits. Definition A.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 . OWARDS HIGHER TOPOLOGY 17
Definition A.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 ∗ Definition A.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 A.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 A.1.6 (Bounded Ionads) . A ionad X is bounded if O ( X ) is a topos.A.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.A.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 – described † in 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 A.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! A.3.2.
We will always omit the adjective generalized . Remark A.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 A.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 A.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 A.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 A.3.7 ([AR18][Cor. 3.8]) . P ( X ) is (finitely) complete if and only if X is (finitely) pre-complete . Corollary A.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. See [AR18][Def. 3.3].
OWARDS HIGHER TOPOLOGY 19
Corollary A.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 A.3.10 (Ionads are generalized ionads) . It is obvious from the previousdiscussion that a ionad is a generalized ionad.
Remark A.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 A.3.12. If f ∗ : G → ¶ ( X ) is a cocontinuous functor from a totalcategory, then it has a right adjoint f ∗ . Remark A.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! A.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 A.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 A.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 A.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 A.3.18.
A ionad X = ( X, Int) is bounded if any of the followingequivalent conditions is verified: 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. † (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 andcall 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 A.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 A.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 PhD thesis. Proposition A.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. OWARDS HIGHER TOPOLOGY 21
C B O ( lan e X e X ) ¶ ( Y ) ¶ ( X ) e Y f ⋄ f ′ e X U X f ∗ (cid:3) Acknowledgements
I am indebted to my advisor, Jiˇr´ı Rosick´y, for the freedom and the trust he blessedme with during these years, not to mention his sharp and remarkably blunt wisdom.I am grateful to Axel Osmond for some very constructive discussions on this topic,and to Richard Garner for several comments that have improved the paper bothtechnically and linguistically. I am grateful to Andrea Gagna for having read andcommented a preliminary draft of this paper.
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