Topological categories, quantaloids and Isbell adjunctions
aa r X i v : . [ m a t h . C T ] J a n Topological categories, quantaloids and Isbell adjunctions
Lili Shen , Walter Tholen ∗ Department of Mathematics and Statistics, York University, Toronto, Ontario, Canada, M3J 1P3
Dedicated to Eva Colebunders on the occasion of her 65th birthday
Abstract
In fairly elementary terms this paper presents, and expands upon, a recent result by Garner bywhich the notion of topologicity of a concrete functor is subsumed under the concept of totalcocompleteness of enriched category theory. Motivated by some key results of the 1970s, thepaper develops all needed ingredients from the theory of quantaloids in order to place essentialresults of categorical topology into the context of quantaloid-enriched category theory, a field thatpreviously drew its motivation and applications from other domains, such as quantum logic andsheaf theory.
Keywords:
Concrete category, topological category, enriched category, quantaloid, totalcategory, Isbell adjunction
1. Introduction
Garner’s [8] recent discovery that the fundamental notion of topologicity of a concrete functormay be interpreted as precisely total (co)completeness for categories enriched in a free quantaloidreconciles two lines of research that for decades had been perceived by researchers in the re-spectively fields as almost intrinsically separate, occasionally with divisive arguments. While thelatter notion is rooted in Eilenberg’s and Kelly’s enriched category theory (see [7, 16]) and theseminal paper by Street and Walters [24] on totality, the former notion goes back to Br¨ummer’sthesis [4] and the pivotal papers by Wyler [31, 32] and Herrlich [10] that led to the developmentof categorical topology and the categorical exploration of a multitude of new structures; see, forexample, the survey by Colebunders and Lowen [18]. The purpose of this paper is to present, andexpand upon, Garner’s result in the most accessible terms for readers who may not necessarily befamiliar with the extensive apparatus of enriched category theory and, in particular, the theory ofquantaloid-enriched categories, as developed mostly in Stubbe’s papers [25, 26].Given a (potentially large) family of objects X i ( i ∈ I ) in a concrete category E over a fixedcategory B (which most often is simply the category of sets) and of maps | X i | / / Y in B (with | X i | denoting the underlying B -object of X i ), topologicity of the functor | - | : E / / B precisely asksfor the existence of a “best” E -structure on Y , called a final lifting of the given structured sink ofmaps. Such liftings are needed not just for the formation of, say, topological sums and quotients (orin the dual situation, of products and subspaces), but in fact belong to the topologist’s standardarsenal when defining new spaces from old in many situations, under varying terminology, such asthat of a“weak topology”. By contrast, at first sight, total cocompleteness appears to be a muchmore esoteric notion, as it entails a very strong existence requirement for colimits of possibly largediagrams, in both ordinary and enriched category theory. The surprising interpretation of final ∗ Corresponding author.
Email addresses: [email protected] (Lili Shen), [email protected] (Walter Tholen) Partial financial assistance by the Natural Sciences and Engineering Research Council of Canada is gratefullyacknowledged. c (cid:13) http://creativecommons.org/licenses/by-nc-nd/4.0/ . The published version is available at http://dx.doi.org/10.1016/j.topol.2015.12.020 . iftings as (so-called weighted) colimits lies at the heart of Garner’s discovery. It was made possibleby his quite simple observation that concrete categories over B may be seen as categories enrichedover the free quantaloid generated by B (see Rosenthal [20]), with the categories enriched in suchbicategories being first defined by Walters [30].Without assuming any a-priori background by the reader on quantaloids, we show in Section2 how concrete categories over B naturally lead to the formation of the free quantaloid over B andtheir interpretation as categories enriched in that quantaloid. Section 3 shows how a given struc-tured sink may, without loss of generality, always be assumed to be a presheaf, which then producesGarner’s result immediately. In Section 4 we discuss quantaloidal generalizations of Wyler’s [32]approach to topological functors, presented as simultaneous fibrations and cofibrations with large-complete fibres, and we carefully compare these notions with their enriched counterparts, namelythat of being tensored, cotensored and conically complete, adding new facts and counter-examplesto the known theory. Sections 5 to 7 give a quick tour of the basic elements of quantaloid-enrichedcategory theory as needed for the presentation of the self-dual concept of totality which, whenapplied in the concrete context, reproduces Hoffmann’s [12] self-duality result for topological func-tors. A key tool here is provided by B´enabou’s distributors [3] which, roughly speaking, generalizefunctors in the same way as relations generalize maps. Based on the paper by Shen and Zhang[23], in Sections 8 and 9 we show that a category is total precisely when it appears as the categoryof the fixed objects under the so-called Isbell adjunction induced by a distributor — which, amongother things, reproduces the MacNeille completion of an ordered set. We also extend the known[27] characterization of totality as injectivity w.r.t. fully faithful functors from small quantaloid-enriched categories to large ones (Section 10). When applied to concrete functors, it reproducesthe characterization of topologicity through diagonal conditions, as first considered in Huˇsek [15],completed in Br¨ummer-Hoffmann [5], and generalized in Tholen-Wischnewsky [29].In this paper we employ no specific strict set-theoretical regime, distinguishing only betweensets (“small”) and classes (“(potentially) large”) and adding the prefix “meta” to categories whoseobjects may be large, thus trusting that our setting may be accommodated within the reader’sfavourite foundational framework. In fact, often the formation of such metacategories may beavoided as it is undertaken only for notational convenience.We thank the anonymous referee for several helpful remarks.
2. Concrete categories as free-quantaloid-enriched categories
For a (“base”) category B with small hom-sets, a category E that comes equipped with afaithful functor | - | : E / / B is usually called concrete (over B ) [1]. Referring to arrows in B as maps , we may then call amap f : | X | / / | Y | with X, Y ∈ ob E a morphism (or more precisely, an E -morphism ) if there is f ′ : X / / Y in E with | f ′ | = f . In other words, since E ( X, Y ) ∼ = |E ( X, Y ) | ⊆ B ( | X | , | Y | ) , being a morphism is a property of maps between E -objects. It is therefore natural to associatewith B a new category Q B , the objects of which are those of B , but the arrows in Q B are sets ofmaps: Q B ( S, T ) = { f | f ⊆ B ( S, T ) } for S, T ∈ ob B . With g ◦ f = { g ◦ f | f ∈ f , g ∈ g } , S = { S } (for f : S / / T , g : T / / U ), Q B becomes a category and, in fact, a quantaloid , i.e., a categorywhose hom-sets are complete lattices such that the composition preserves suprema in each variable.Indeed, as is well known, Q B is the free quantaloid generated by B :2 roposition 2.1. [20] The assignment
B 7→ Q B defines a left adjoint to the forgetful 2-functor QUANT / / CAT , which forgets the ordered structure of a quantaloid.
A category E concrete over B may now be completely described by • a class ob E of objects; • a function | - | : ob E / / ob Q B (= ob B ) sending each object X in E to its extent | X | in B ; • a family E ( X, Y ) ∈ Q B ( | X | , | Y | ) ( X, Y ∈ ob E ), subject to – 1 | X | ⊆ E ( X, X ), – E ( Y, Z ) ◦ E ( X, Y ) ⊆ E ( X, Z ) (
X, Y, Z ∈ ob E ).A (concrete) functor F : E / / D between concrete categories over B (that must commute withthe respective faithful functors to B ) may then be described by a function F : ob E / / ob D with • | F X | = | X | , • E ( X, Y ) ⊆ D ( F X, F Y ) (
X, Y ∈ ob E ).An order between concrete functors F, G : E / / D is given by F ≤ G ⇐⇒ ∀ X ∈ ob E : 1 | X | : | F X | / / | GX | is a D -morphism ⇐⇒ ∀ X ∈ ob E : | X | ⊆ D ( F X, GX ) , rendering the (meta)category CAT ⇓ c B of concrete categories over B as a 2-category, with 2-cells given by order.Above we have described categories and functors concrete over B as categories and functors enriched in Q B . In fact, for any quantaloid Q , a Q -category E may be defined precisely as above,by just trading Q B for Q and “ ⊆ ” for the order “ ≤ ” of the hom-sets of Q , and likewise for a Q -functor F : E / / D and the order of Q -functors. With Q - CAT denoting the resulting 2-(meta)category of Q -categories and Q -functors one may now formally confirm the equivalence ofthe descriptions given above, as follows: Proposition 2.2. [8]
CAT ⇓ c B and Q B - CAT are 2-equivalent.
In what follows, for a concrete category E over B , we write E for the corresponding Q B -category.Hence, ob E = ob E and E ( X, Y ) := |E ( X, Y ) | ⊆ B ( | X | , | Y | )for all X, Y ∈ ob E .Let us also fix the notation for the right adjoints of the join preserving functions − ◦ u : Q ( T, U ) / / Q ( S, U ) and v ◦ − : Q ( S, T ) / / Q ( S, U )for u : S / / T and v : T / / U , respectively, in any quantaloid Q . They are defined such that theequivalences v ≤ w ւ u ⇐⇒ v ◦ u ≤ w ⇐⇒ u ≤ v ց w hold for all u , v as above, and w : S / / U in Q , i.e., w ւ u = _ { v ∈ Q ( T, U ) | v ◦ u ≤ w } and v ց w = _ { u ∈ Q ( S, T ) | v ◦ u ≤ w } . For Q = Q B and f ⊆ B ( S, T ), g ⊆ B ( T, U ) and h ⊆ B ( S, U ), these formulas give h ւ f = { g ∈ B ( T, U ) | ∀ f ∈ f : g ◦ f ∈ h } and g ց h = { f ∈ B ( S, T ) | ∀ g ∈ g : g ◦ f ∈ h } . In this paper, “order” refers to a reflexive and transitive relation (usually called preorder), with no requirementfor antisymmetry, unless explicitly stated. . Structured sinks as presheaves, topological functors as total categories For a category E concrete over B and an object T in B , a structured sink σ over T is givenby a (possibly large) family of objects X i in E and maps f i : | X i | / / T , i ∈ I . A lifting of σ = ( T, X i , f i ) i ∈ I is an E -object Y with | Y | = T such that all maps f i are E -morphisms, and thelifting is final (w.r.t. | - | : E / / B ) if any map g : | Y | / / | Z | becomes an E -morphism as soon asall maps g ◦ f i : | X i | / / | Z | are E -morphisms. The finality property means precisely E ( Y, Z ) = { g ∈ B ( | Y | , | Z | ) | ∀ i ∈ I : g ◦ f i ∈ E ( X i , Z ) } = \ i ∈ I E ( X i , Z ) ւ { f i } for all Z ∈ ob E .Replacing the singleton sets { f i } by subsets f i ⊆ B ( | X i | , T ), we may assume that the objects X i are pairwise distinct. In fact, since f i is allowed to be empty, without loss of generality, wemay always assume the indexing class of σ to be ob E . So, our structured sink σ is now describedas an ob E -indexed family of subsets σ X ⊆ B ( | X | , T ), and its final lifting Y is characterized by E ( Y, Z ) = \ X ∈ ob E E ( X, Z ) ւ σ X ( ∗ )for all Z ∈ ob E .Moreover, since for all X, Z ∈ ob E , one trivially has E ( X, Z ) ւ σ X = \ X ′ ∈ ob E E ( X ′ , Z ) ւ ( σ X ◦ E ( X ′ , X )) , we may assume σ to be closed under composition with E -morphisms from the right. That is, σ X ◦ E ( X ′ , X ) ⊆ σ X ′ ( ∗∗ )for all X, X ′ ∈ ob E .Recall that a faithful functor | - | : E / / B is topological if all structured sinks admit final liftings.The above considerations show: Proposition 3.1.
A concrete category E is topological over B if, for all families σ = ( T, σ X ) X ∈ ob E with σ X ⊆ B ( | X | , T ) satisfying ( ∗∗ ), there is Y ∈ ob E satisfying ( ∗ ). In the terminology of quantaloid-enriched categories (see [21, 23, 25]), topologicity is thereforecharacterized by the existence of suprema of all presheaves. Indeed, for any quantaloid Q , a presheaf ϕ on a Q -category E of extent T is given by a family of arrows ϕ X : | X | → T in Q with ϕ X ◦ E ( X ′ , X ) ≤ ϕ X ′ for all X, X ′ ∈ ob E . A supremum of ϕ is an object Y = sup ϕ in E with extent T satisfying ( ∗ ) transformed to the current context: E (sup ϕ, Z ) = ^ X ∈ ob E E ( X, Z ) ւ ϕ X ( ∗ ′ )for all Z ∈ ob E .This latter condition is expressed more compactly once the presheaves on E have been organizedas a (very large) Q -category P E , with hom-arrows P E ( ϕ, ψ ) = ^ X ∈ ob E ψ X ւ ϕ X .
4n fact, choosing for ψ the presheaf E ( − , Z ) of extent | Z | , condition ( ∗ ′ ) reads as E (sup ϕ, Z ) = P E ( ϕ, E ( − , Z )) ( ∗ ′′ )for all Z ∈ ob E . In terms of the Yoneda Q -functor Y E : E / / P E , Z
7→ E ( − , Z ) , this condition reads as E (sup ϕ, Z ) = P E ( ϕ, Y Z ) ( ∗ ′′′ )for all Z ∈ ob E , ϕ ∈ ob P E . In other words, with the usual adjointness terminology transferred tothe Q -context, one obtains: Theorem 3.2 (Garner [8]) . A concrete category E over B is topological if, and only if, the Q B -category E is total; that is, if the Yoneda Q B -functor Y E has a left adjoint. For general Q , a Q -category E is called total if the Yoneda Q -functor Y E has a left adjoint.Total Q -categories will be studied further beginning from Section 5.
4. Cofibred categories as tensored categories
Recall that a faithful functor | - | : E / / B is a cofibration if every structured singleton-sink hasa final lifting; that is, if for every map f : | X | / / T with X ∈ ob E there is some Y ∈ ob E with E ( Y, Z ) = E ( X, Z ) ւ { f } for all Z ∈ ob E . We write Y = f ⋆ X and call E cofibred (over B ) in this case. Dually, E is fibred (over B ) if | - | op : E op / / B op is a cofibration.Denoting by E T the fibre of | - | at T ∈ ob B , which is a possibly large ordered class, one has thefollowing well-known characterization of topologicity of concrete categories (see [28], [14, TheoremII.5.9.1]): Theorem 4.1.
A concrete category E over B is topological if, and only if, it is fibred and cofibredand its fibres are large-complete ordered classes. A proof of this theorem appears below as Corollary 6.3.Wyler [32] originally introduced topological categories using this characterization, which im-plies in particular the self-duality of topologicity, but restricting himself to concrete categorieswith small fibres. Without that restriction, self-duality (so that E is topological over B if and onlyif E op is topological over B op ) was first established by Hoffmann [12, 13].Although Theorem 4.1 has a natural generalization to the context of quantaloid-enriched cat-egories (which will be discussed in the next section), there is no immediate “translation” of thenotion of (co)fibration into that context. If, however, we require every Q B -arrow f : | X | / / T with X ∈ ob E (instead of just a single map f in B ) to have a final lifting Y ∈ ob E , so that E ( Y, Z ) = E ( X, Z ) ւ f ( † )for all Z ∈ ob E , then we do obtain a notion which is familiar for quantaloid-enriched categories,as we recall next.For any quantaloid Q , a Q -category E is tensored [23, 26] if for all X ∈ ob E and u : | X | / / T in Q , there is some Y ∈ ob E with | Y | = T and E ( Y, Z ) = E ( X, Z ) ւ u ( † ′ )for all Z ∈ ob E . Such object Y , also called the tensor of u and X , we denote by u ⋆ X . Theterminology and notation deserves some explanation.5irst of all, for every object S in Q , one has the Q -category { S } with only one object S withextent S , and hom-arrow { S } ( S, S ) = 1 S . The presheaf Q -category P { S } has arrows u : S / / T in Q as its objects, and their extent is their codomain: | u | = T ; the hom-arrows are given by P { S } ( u, v ) = v ւ u = _ { w ∈ Q ( T, U ) | w ◦ u ≤ v } where v : S / / U . Now, for every object X of a Q -category E , there is a Q -functor E ( X, − ) : E / / P { S } with S = | X | , and we can rewrite ( † ′ ) as E ( u ⋆ X, Z ) = P { S } ( u, E ( X, Z )) ( † ′′ )for all Z ∈ ob E . This shows instantly: Proposition 4.2 (Stubbe [26]) . A Q -category E is tensored if, and only if, for all objects X in E , the Q -functor E ( X, − ) : E / / P {| X |} has a left adjoint (given by tensor). As explained above, for the Q B -category E of a concrete category E over B to be tensored, itis necessary that E be cofibred over B . Here is the precise differentiation of the two properties inquestion: Proposition 4.3.
Let E be a category concrete over B . Then the Q B -category E is tensored if, andonly if, E is cofibred with the additional property that for all X ∈ ob E , T ∈ ob B and f ⊆ B ( | X | , T ) ,there is Y ∈ ob E with | Y | = T and E ( Y, Z ) = \ f ∈ f E ( f ⋆ X, Z ) ( ‡ ) for all Z ∈ ob E .Proof. It suffices to show that the right-hand sides of ( † ) and ( ‡ ) coincide. But for all g ∈ B ( T, | Z | ),one has g ∈ E ( X, Z ) ւ f ⇐⇒ ∀ f ∈ f : g ◦ f is an E -morphism X / / Z ⇐⇒ ∀ f ∈ f : g is an E -morphism f ⋆ X / / Z ⇐⇒ g ∈ \ f ∈ f E ( f ⋆ X, Z ) . Corollary 4.4.
For a non-empty category E concrete over B with E tensored, the functor | - | : E / / B has a fully faithful left adjoint and right inverse functor.Proof. Independently of the chosen X in E , for f = ∅ condition ( ‡ ) reads as E ( Y, Z ) = B ( T, | Z | )for all Z ∈ ob E , exhibiting Y as the value of the left adjoint at T . Remark 4.5.
For a cofibred category E over B , the Q B -category E may fail to be tensored; in fact, | - | : E / / B may fail to have a fully faithful left adjoint. Indeed, for any category B that is notjust an ordered class (so that there is at least one hom-set B ( T, Z ) with at least two morphisms),the codomain functor ( T ↓ B ) / / B
6f the comma category of B under T is cofibred over B but does not have a fully faithful leftadjoint (at T ). Otherwise there would be a map j : T / / T such that 1 T : T / / cod j would serveas the adjunction unit at T , i.e., B ( T, cod g ) = ( T ↓ B )( j, g )for all g : T / / Z . Considering first g = 1 T one sees that one must have also j = 1 T , and thenthat B ( T, Z ) can contain only at most one map.
Remark 4.6.
In the order of the fibre E T of the functor | - | : E / / B over T ∈ ob B , an object Y satisfying ( ‡ ) is necessarily the join of the objects ( f ⋆ X ) f ∈ f : just constrain the objects Z torange in E T . Hence, when E is tensored, the fibres of | - | admit certain small-indexed joins.However, for a cofibred category E over B whose fibres are complete, E may still fail to betensored, as is shown by the following example. Example 4.7.
Consider the category B of complete lattices, with monotone (but not necessarilyjoin-preserving) functions as maps; and let E be the category of pointed complete lattices, withmorphisms f : ( X, x ) / / ( Y, y ) monotone functions satisfying f ( x ) ≤ y . (Hence, E is a laxversion of the comma category (1 ↓ B ).) Then E is cofibred over B , and the fibre of the forgetfulfunctor E / / B at any X is isomorphic to the complete lattice X itself. But considering anymonotone function g : Y / / Z in B that fails to preserve joins, so that there are y i ∈ Y , i ∈ I with z = _ i ∈ I g ( y i ) < g ( y ) , y = _ i ∈ I y i , for the constant maps f i : 1 / / Y with value y i we then have ( Y, y i ) = f i ⋆ g ∈ E ( f i ⋆ , ( Z, z ))for all i ∈ I , but g
6∈ E (( Y, y ) , ( Z, z )), even though (
Y, y ) = _ i ∈ I ( Y, y i ) in the fibre E Y . Hence,condition ( ‡ ) of Proposition 4.3 is violated.These remarks underline the difference between order-completeness and conical cocompletenessin quantaloid-enriched categories. Indeed, for a general quantaloid Q , a Q -category E is order-complete [26] if, for all T ∈ ob Q , the class E T of E -objects of extent T ordered by Y ≤ Y ′ ⇐⇒ T ≤ E ( Y, Y ′ )admits all joins (and equivalently, meets). In the presheaf Q -category P E , this order amounts tothe componentwise order inherited from Q : ϕ ≤ ψ ⇐⇒ T ≤ P E ( ϕ, ψ ) ⇐⇒ ∀ X ∈ ob E : 1 T ≤ ψ X ւ ϕ X ⇐⇒ ∀ X ∈ ob E : ϕ X ≤ ψ X . It is therefore clear that P E is order-complete.One calls E conically cocomplete [26] if, for all T ∈ ob Q , joins of representable presheavestaken in ( P E ) T have suprema, that is: if for any (possibly large) family of objects Y i ∈ E T , i ∈ I ,the supremum Y of the presheaf ϕ = _ i ∈ I E ( − , Y i )exists. Such Y must necessarily satisfy Y = _ i ∈ I Y i in E T . Indeed, the restriction of condition ( ∗ ′ )(see Section 3) yields Y ≤ Z ⇐⇒ ∀ X ∈ ob E : 1 T ≤ E ( X, Z ) ւ ϕ X ⇐⇒ ∀ X ∈ ob E : ϕ X ≤ E ( X, Z ) ⇐⇒ ∀ X ∈ ob E , ∀ i ∈ I : E ( X, Y i ) ≤ E ( X, Z ) ⇐⇒ ∀ i ∈ I : Y i ≤ Z. orollary 4.8. If the concrete category E over B is cofibred with E conically cocomplete, then E is tensored.Proof. We show that for all X ∈ ob E , T ∈ ob B and f ⊆ B ( | X | , T ), an object Y in E with | Y | = T satisfying ( ‡ ) in Proposition 4.3 is exactly the conical colimit of the objects f ⋆ X , f ∈ f . It sufficesto show that the right-hand sides of ( ‡ ) and ( ∗ ) in Section 3 coincide for σ = [ f ∈ f E ( − , f ⋆ X ).Indeed, for all Z ∈ ob E , g : T / / Z in B , g ∈ \ f ∈ f E ( f ⋆ X, Z ) ⇐⇒ ∀ f ∈ f : g is an E -morphism f ⋆ X / / Z ⇐⇒ ∀ f ∈ f , W ∈ ob E , h ∈ E ( W, f ⋆ X ) : g ◦ h is an E -morphism W / / Z ⇐⇒ ∀ W ∈ ob E , h ∈ [ f ∈ f E ( W, f ⋆ X ) : g ◦ h is an E -morphism W / / Z ⇐⇒ g ∈ \ W ∈ ob E (cid:16) E ( W, Z ) ւ [ f ∈ f E ( W, f ⋆ X ) (cid:17) . However, for a concrete category E over B such that E is tensored and order-complete, E maystill fail to be conically cocomplete, as is shown by the following example: Example 4.9.
Let the category B have only one object with exactly two endomorphisms i, e ,the non-identity morphism e being idempotent. The objects of the category E are the naturalnumbers, plus a largest element ∞ adjoined, and its hom-sets are given by E ( X, Y ) = { i, e } , if X ≤ Y, { e } , if Y < X < ∞ , ∅ , if X = ∞ , Y < ∞ . With composition as in B and | - | : E / / B mapping morphisms identically, E is concrete over B . Furthermore, E is tensored since, for all X ∈ ob E , E ( X, − ) preserves meets (thus has a leftadjoint). But E is not conically cocomplete since ϕ = _ X< ∞ E ( − , X ) does not have a supremum.Indeed, the supremum would have to be the join ∞ of the objects X < ∞ ; but for any Z < ∞ one has \ Y ∈ ob E E ( Y, Z ) ւ ϕ Y = { e } while E ( ∞ , Z ) = ∅ .What is needed to make order-completeness equivalent to conical cocompleteness is fibredness: Proposition 4.10.
For a fibred category E over B , the Q B -category E is conically cocompletewhenever it is order-complete.Proof. Let Y = _ i ∈ I Y i in E T , for T ∈ ob B . With ϕ = _ i ∈ I E ( − , Y i ) we must show E ( Y, Z ) = \ X ∈ ob E E ( X, Z ) ւ ϕ X for all Z ∈ ob E . But for a map g : T / / | Z | in the right-hand side set and e Y := ( g Z ) the( | - | )-initial lifting of g one has Y i ≤ e Y for every i ∈ I (consider X = Y i ). Consequently, Y ≤ e Y ,and therefore g : | Y | / / | Z | must be an E -morphism. This proves “ ⊇ ”, the other inclusion beingtrivial. 8 emark 4.11. We note, however, that a conically cocomplete concrete category need not befibred. Indeed, similarly to Example 4.7 consider for B the category of complete lattices withjoin-preserving maps, and let E be pointed complete lattices with morphisms preserving joinsstrictly but base-points only laxly. Then E is obviously not fibred over B although E is conicallycocomplete.
5. Distributors, weighted colimits, total cocompleteness
Suprema of presheaves (as used in Section 3) are special weighted colimits that we shouldmention here in full generality. For this, in turn, it is convenient to have the language of distributorsat one’s disposal. For a quantaloid Q , a Q -distributor (also bimodule or profunctor ) Φ : E / / ◦ D of Q -categories E , D is given by a family of arrows Φ( X, Y ) : | X | / / | Y | in Q ( X ∈ ob E , Y ∈ ob D )in Q such that D ( Y, Y ′ ) ◦ Φ( X, Y ) ◦ E ( X ′ , X ) ≤ Φ( X ′ , Y ′ )( X, X ′ ∈ ob E , Y, Y ′ ∈ ob D ). Every Q -category E may be considered as a Q -distributor E : E / / ◦ E and, in fact, serves as an identity Q -distributor when one defines the composite of Φ followed byΨ : D / / ◦ C via (Ψ ◦ Φ)(
X, Z ) = _ Y ∈ ob D Ψ( Y, Z ) ◦ Φ( X, Y ) . With the pointwise order inherited from Q , we obtain the 2-(meta)category Q - DIS of Q -categories and their Q -distributors. Q - DIS is in fact a (very large) quantaloid, i.e., enrichedover
SUP , the (meta)category of large-complete ordered classes and sup-preserving functions.Every Q -functor F : E / / D gives rise to the Q -distributors F ♮ : E / / ◦ D , F ♮ ( X, Y ) = D ( F X, Y ) ,F ♮ : D / / ◦ E , F ♮ ( Y, X ) = D ( Y, F X ) , so that one has 2-functors( − ) ♮ : ( Q - CAT ) co / / Q - DIS , ( − ) ♮ : ( Q - CAT ) op / / Q - DIS , which map objects identically. Here “co” refers to the dualization of 2-cells; while ( − ) ♮ is covarianton 1-cells but inverts their order, ( − ) ♮ is contravariant on 1-cells but keeps their order: F ≤ F ′ ⇐⇒ ∀ X ∈ ob E : 1 | X | ∈ D ( F X, F ′ X ) ⇐⇒ ∀ X, Y ∈ ob E : D ( F X, Y ) ≥ D ( F ′ X, Y ) ⇐⇒ F ♮ ≥ ( F ′ ) ♮ ⇐⇒ ∀ X, Y ∈ ob E : D ( Y, F X ) ≤ D ( Y, F ′ X ) ⇐⇒ F ♮ ≤ ( F ′ ) ♮ . Since
E ≤ F ♮ ◦ F ♮ and F ♮ ◦ F ♮ ≤ D , one has F ♮ ⊣ F ♮ in Q - DIS , an important fact that weexploit next.For a Q -category E , every presheaf ϕ ∈ P E may be considered as a Q -distributor ϕ : E / / ◦ {| ϕ |} when one writes ϕ ( X, | ϕ | ) for ϕ X . Every Q -functor F : E / / D now gives the Q -functor F ∗ : P D / / P E , ψ ψ ◦ F ♮ , so that ( F ∗ ψ ) X = ( ψ ◦ F ♮ )( X, | ψ | ) = ψ ( F X, | ψ | ) = ψ F X ( X ∈ ob E ) . The following lemma is well known: 9 emma 5.1. F ∗ has a left adjoint F ! , given by ( F ! ϕ ) Y = _ X ∈ ob E ϕ X ◦ D ( Y, F X ) . Proof.
The given formula translates to F ! ϕ = ϕ ◦ F ♮ . Consequently, F ! ⊣ F ∗ follows easily from F ♮ ⊣ F ♮ .Let D : J / / E be a Q -functor (considered as a “diagram” in E ). For ϕ ∈ P J , a weightedcolimit of D by ϕ is an object Y in E with | Y | = | ϕ | and E ( Y, Z ) = P J ( ϕ, E ( D − , Z ))for all Z ∈ ob E ; one writes Y = ϕ ⋆ D in this case. Here E ( D − , Z ) is the value of the composite Q -functors E Y E / / P E D ∗ / / P J at Z .Since D ! ⊣ D ∗ , so that P E ( D ! ϕ, E ( − , Z )) = P J ( ϕ, D ∗ Y E Z ), the weighted colimit ϕ ⋆ D existsprecisely when sup D ! ϕ exists, and then ϕ ⋆ D ∼ = sup D ! ϕ. Remark 5.2.
The supremum of ϕ ∈ ob P E is precisely the weighted colimit of the identity Q -functor of E by ϕ : sup ϕ = ϕ ⋆ E . The tensor of X ∈ ob E and u : | X | / / T in Q is precisely theweighted colimit of {| X |} / / E , | X | 7→ X , by u : u ⋆ X = u ⋆ ( {| X |} / / E ).A Q -category E is totally cocomplete if ϕ ⋆ D exists for any diagram D in E and weight ϕ ;equivalently, if for every Q -functor D with codomain E , the composite Q -functor D ∗ Y E has a leftadjoint. This is certainly the case when E is total, so that Y E has a left adjoint, since D ∗ hasalways a left adjoint, by Lemma 5.1. More comprehensively, we may now state: Theorem 5.3 (Stubbe [25]) . The following are equivalent for a Q -category E : (i) E is total; (ii) E is totally cocomplete; (iii) E has all suprema; (iv) E is tensored and conically cocomplete.Proof. (i) = ⇒ (ii): See above.(ii) = ⇒ (iii) & (iv): By Remark 5.2.(iii) = ⇒ (i): By definition of totality.(iv) = ⇒ (iii): Given ϕ ∈ ob P E , since E is tensored, for every X ∈ ob E one has Y X = ϕ X ⋆ X with | Y X | = | ϕ | and, since E is conically cocomplete, there is Y = sup ψ with ψ = _ X ∈ ob E E ( − , Y X ) . Z ∈ ob E , the following calculation is easily validated: E ( Y, Z ) = ^ W ∈ ob E E ( W, Z ) ւ ψ W = ^ W ∈ ob E ^ X ∈ ob E E ( W, Z ) ւ E ( W, Y X )= ^ X ∈ ob E ^ W ∈ ob E E ( W, Z ) ւ E ( W, Y X )= ^ X ∈ ob E E ( Y X , Z )= ^ X ∈ ob E E ( X, Z ) ւ ϕ X . Consequently, Y = sup ϕ . Corollary 5.4.
For a concrete category E over B , the following are equivalent: (i) E is topological over B ; (ii) the Q B -category E is tensored and conically cocomplete; (iii) E is cofibred over B , and E is conically cocomplete.Proof. (i) ⇐⇒ (ii) follows from Theorems 3.2 and 5.3, and (ii) ⇐⇒ (iii) follows from Corollary4.8.
6. Dualization
Let us now show how the self-duality of topologicity (as stated in Theorem 4.1) plays itself outfor general Q -categories. First of all, for any quantaloid Q , dualized as an ordinary category, Q op is a quantaloid again, with Q op ( T, S ) = Q ( S, T ) carrying the same order. Every Q -category E induces the Q op -category E op with E op ( Y, X ) = E ( X, Y ), and a Q -functor F : E / / D becomes a Q op -functor F op : E op / / D op . But when F ≤ F ′ for F ′ : E / / D , one has ( F ′ ) op ≤ F op . Briefly,there is a 2-isomorphism ( − ) op : ( Q - CAT ) co / / Q op - CAT . One can now dualize the constructions and notions encountered thus far, as follows: • P † E := ( P ( E op )) op (the covariant presheaf category of E , as opposed to the contravariant presheaf category P E ); • inf E ϕ := sup E op ϕ (the infimum of ϕ ∈ P † E ); • ϕ D := ϕ ⋆ D op (the weighted limit of D : J / / E by ϕ ∈ P † J ); • Y †E := ( Y E op ) op : E / / P † E , X
7→ E ( X, − ) (the dual Yoneda Q -functor); • E cototal : ⇐⇒ E op total ⇐⇒ Y †E has a right adjoint; • E totally complete : ⇐⇒ E op totally cocomplete ⇐⇒ E has all weighted limits; • E cotensored : ⇐⇒ E op tensored; • E conically complete : ⇐⇒ E op conically cocomplete.For our next steps, it is convenient to have a Q -version of the Adjoint Functor Theorem at ourdisposal, as follows: 11 roposition 6.1.
Let E be tensored and order-complete. Then a Q -functor F : E / / D has aright adjoint if, and only if, F preserves tensors and the restrictions F T : E T / / D T ( T ∈ ob Q ) of F to the fibres preserve arbitrary joins.Proof. If F has a right adjoint G , then F preserves all existing weighted colimits, in particulartensors, and since F ⊣ G implies F T ⊣ G T for all T ∈ ob Q , F preserves also all joins in the fibres.Conversely, assuming preservation of tensors and joins, one first observes that every F T must havea right adjoint G T since E is order-complete. Putting GY = G | Y | Y, for all X ∈ ob E , Y ∈ ob D one trivially has E ( X, GY ) ≤ D ( F X, F GY ) ≤ D ( F X, Y ) , and from 1 | Y | ≤ E ( X, GY ) ւ E ( X, GY )= E ( E ( X, GY ) ⋆ X, GY ) ( E tensored)= D ( F ( E ( X, GY ) ⋆ X ) , Y ) ( F | Y | ⊣ G | Y | )= D ( E ( X, GY ) ⋆ F X, Y ) ( F preserves tensors)= D ( F X, Y ) ւ E ( X, GY )one obtains D ( F X, Y ) ≤ E ( X, GY ). Hence, F ⊣ G .One can now extend the list of equivalent statements of Theorem 5.3 by its dualizations, asfollows: Theorem 6.2 (Stubbe [25]) . A Q -category E is total if and only if the following equivalent con-ditions hold: (v) E is cototal; (vi) E is totally complete; (vii) E has all infima; (viii) E is cotensored and conically complete.Proof. It suffices to prove that E is cototal when E is total, and thanks to Proposition 6.1 andRemark 5.2, for that it suffices to prove that the dual Yoneda Q -functor Y † : E / / P † E preservesall weighted colimits. Here is a quick sketch of that fact. First note:(a) In the (meta)quantaloid Q - DIS , for Φ : E / / ◦ D , Ψ : D / / ◦ C , Ξ : E / / C one has(Ξ ւ Φ)(
Y, Z ) = ^ X ∈ ob E Ξ( X, Z ) ւ Φ( X, Y ) , (Ψ ց Ξ)(
X, Y ) = ^ Z ∈ ob C Ψ( Y, Z ) ց Ξ( X, Z )for all X ∈ ob E , Y ∈ ob D , Z ∈ ob C .(b) For Φ : E / / ◦ C , Ψ : D / / ◦ B and a Q -functor F : E / / D one has(Ψ ◦ F ♮ ) ւ Φ = Ψ ւ (Φ ◦ F ♮ ) . (c) Y = Y E (and, hence, Y †E ) is fully faithful, that is: Y ♮ ◦ Y ♮ = E .12ow consider the weighted colimits ϕ ⋆ D of D : J / / E by ϕ ∈ ob P J , then Y † ( ϕ ⋆ D ) = E ( ϕ ⋆ D, − ) (definition of Y † )= D ♮ ւ ϕ (definition of weighted colimit, (a))= (( Y † ) ♮ ◦ Y † ♮ ◦ D ♮ ) ւ ϕ (by (c))= (( Y † ) ♮ ◦ ( Y † D ) ♮ ) ւ ϕ (functoriality of ( − ) ♮ )= ( Y † ) ♮ ւ ( ϕ ◦ ( Y † D ) ♮ ) (by (b))= ϕ ⋆ Y † D ;here the last step follows from the fact, that for any Q -functor F : J / / P † E (in lieu of Y † D ), theweighted colimit of F by ϕ may be computed as ϕ ⋆ F = sup P † E F ! ϕ = ( Y † ) ♮ ւ F ! ϕ = ( Y † ) ♮ ւ ( ϕ ◦ F ♮ ) . Corollary 6.3.
For a concrete category E over B , the following assertions are equivalent: (i) E is topological over B ; (ii) E op is topological over B op ; (iii) E is fibred and cofibred over B with large-complete fibres.Proof. Since Q B op = ( Q B ) op and E op = E op , the equivalence of (i) and (ii) follows from Theorems3.2 and 6.2, which also imply (i)&(ii) = ⇒ (iii). The converse implication follows with Corollary4.8 and Proposition 4.10.
7. Universality of the presheaf construction
First, let us briefly recall the
Yoneda Lemma for Q -categories: Lemma 7.1.
For a Q -category E and all X ∈ ob E , ϕ ∈ ob P E , one has P E ( Y E X, ϕ ) = ϕ X . As a consequence one obtains the following fundamental adjunction:
Proposition 7.2.
For Q -categories E , C , there is a natural 1-1 correspondence E C Ψ / / ◦C P E G / / ( G ♮ ◦ ( Y E ) ♮ = Ψ) , which respects the order of Q -functors and Q -distributors.Proof. Given Ψ, a Q -functor G with G ♮ ◦ ( Y E ) ♮ = Ψ must necessarily satisfy( GZ ) X = P E ( Y E X, GZ ) = Ψ(
X, Z )for all Z ∈ ob C , X ∈ ob E . Conversely, defining G in this way one obtains a Q -functor.When restricting ourselves to the case of a small quantaloid Q and to considering small Q -categories, we therefore obtain an adjunction( Q - Cat ) op Q - Dis . o o P ( Q - Cat ) op Q - Dis . ( − ) ♮ / / ⊥ − ) ♮ maps objects identically while the presheaf functor P assigns to a Q -distributor Φ : E / / ◦ D the Q -functor Φ ∗ : P D / / P E with (Φ ∗ ) ♮ ◦ ( Y E ) ♮ = ( Y D ) ♮ ◦ Φ, that is:(Φ ∗ ψ ) X = _ Z ∈ ob D ψ Z ◦ Φ( X, Z )for all ψ ∈ ob P D , X ∈ ob E . The unit of the adjunction at E is ( Y E ) ♮ while Y E is the counit (since( Y E ) ♮ ◦ ( Y E ) ♮ = E ). Note that, for a Q -functor F : E / / D , one has( F ♮ ) ∗ = F ∗ and ( F ♮ ) ∗ = F ! with F ∗ , F ! defined as in Lemma 5.1. Consequently, there is a monad( P , Y , S )on the 2-category Q - Cat , with P mapping F to F ! and with the monad multiplication S given by S E Φ = _ ϕ ∈ ob E Φ ϕ ◦ ϕ X for all Φ ∈ PP E , X ∈ ob E . It is straightforward to show that this monad is of Kock-Z¨oberleintype , that is, that ( Y E ) ! = ( Y ♮ E ) ∗ ≤ Y P E for every Q -category E , and consequently S E ⊣ Y P E : P E / / PP E . In other words, every Φ ∈ PP E has a supremum: sup Φ = S E Φ. This fact of course remains truealso for large Q and E ; one just has to accept the fact that P E and PP E will generally live inhigher universes than E . Corollary 7.3.
Yoneda maps every Q -category E fully and faithfully into the total Q -(meta)category P E . Theorem 7.4.
For all Q -categories E , D with D total, there is a natural 1-1 correspondence E D F / / P E D , H preserves weighted colimits H / / ( H Y E = F ) which respects the order of Q -functors.Proof. Let us first note that every ϕ ∈ ob P E is the weighted colimit of Y E by ϕ , since P E ( ϕ, ψ ) = P E ( ϕ, P E ( Y E − , ψ ))for all ψ ∈ ob P E . Hence, given F , any H with H Y E = F that preserves all weighted colimits mustsatisfy Hϕ = H ( ϕ ⋆ Y E ) = ϕ ⋆ H Y E = ϕ ⋆ F. Conversely, a straightforward computation shows that H defined in this way is actually thecomposite Q -functor P E F ! / / P D sup D / / D which, as the composite of two left adjoints, must preserve all weighted colimits. Furthermore, H Y E = sup D F ! Y E = sup D Y D F = F since Y D is fully faithful. 14onsequently, for small Q there is an adjunction Q - TotCat Q - Cat , o o P Q - TotCat Q - Cat , / / ⊥ with Q - TotCat denoting the category of small total Q -categories and their weighted-colimit-preserving Q -functors. The induced monad on Q - Cat is again ( P , Y , S ), as described beforeCorollary 7.3. Corollary 7.5 (Herrlich [10]) . For a concrete category over B , the topological (meta)category P E over B has the following universal property: Every concrete functor F : E / / D into a topologicalcategory D over B factors uniquely through a concrete functor H : P E / / D with H Y E = F thatpreserves final sinks. Here the objects of P E are structured sinks satisfying the closure property ( ∗∗ ) of Section 3,and the full concrete embedding Y E assigns to every object X the structured sink of all maps withcodomain | X | . Proof.
The only point to observe is that a concrete functor preserves final sinks if and only if itpreserves suprema (see Section 3) or, equivalently, weighted colimits (see Section 5).As a consequence, the category of small topological categories over the small category B andthe finality-preserving concrete functors admits a right adjoint forgetful functor into Cat ⇓ c B .
8. Total Q -categories are induced by Isbell adjunctions Let us re-interpret Proposition 7.2, by setting up the (meta-)2-category Q - CHU whose objects are given by Q -distributors, and whose morphisms( F, G ) : Φ / / Ψare given by Q -functors F : E / / D , G : C / / B which make the diagram B C G ♮ / / EB Φ (cid:15) (cid:15) E D F ♮ / / DC Ψ (cid:15) (cid:15) ◦◦◦ ◦ commute or, equivalently, satisfy the diagonal condition Ψ ւ F ♮ = G ♮ ց Φ . With composition and order defined as in Q - CAT , Q - CHU becomes a (meta-)2-category whosemorphisms may also be referred to as Q -Chu transforms (in generalization of the terminology usedfor morphisms of Chu spaces; see [2, 19, 9]). There is an obvious 2-functordom : Q - CHU / / Q - CAT , ( F, G ) F. Proposition 8.1.
For all Q -categories E and Q -distributors Ψ : D / / C , there is a natural 1-1correspondence E dom Ψ F / / ( Y E ) ♮ Ψ ( F,G ) / / which preserves the order of Q -functors and Q -Chu transforms. roof. Given F , by Proposition 7.2 there is a unique Q -functor G : C / / P E with G ♮ ◦ ( Y E ) ♮ = Ψ ◦ F ♮ ,that is: with ( F, G ) : ( Y E ) ♮ / / Ψ a Q -Chu transform.Under the restriction to small Q -categories (for Q small as well) we therefore obtain a full andfaithful left adjoint to dom : Q - Chu / / Q - Cat . Corollary 8.2.
The assignment
E 7→ ( Y E ) ♮ embeds Q - Cat into Q - Chu as a full coreflectivesubcategory.
Remark 8.3.
Note that the embedding of Q - Cat / / Q - Chu of Corollary 8.2 does NOT preservethe local order in Q - Cat , i.e., it is not a 2-categorical embedding.Every Q -distributor Φ : E / / ◦ B induces the Isbell adjunction [21, 23] (in generalization of theterminology introduced by Lawvere [17] for enriched categories) P E P † B , Φ ↑ / / P E P † B , o o Φ ↓ ⊥ ϕ Φ ւ ϕ ✤ / / ψ ց Φ ψ o o ✤ ⊥ . Indeed, the easily established identity ψ ց (Φ ւ ϕ ) = ( ψ ց Φ) ւ ϕ translates to P † B (Φ ↑ ϕ, ψ ) = P E ( ϕ, Φ ↓ ψ )for all ϕ ∈ ob P E , ψ ∈ ob P B . We denote by I Φ the (very large) full reflective Q -subcategory of P E of presheaves fixed by the adjunction, i.e.,ob( I Φ) = { ϕ ∈ ob P E | Φ ↓ Φ ↑ ϕ = ϕ } , and call I Φ the
Isbell Q -category of Φ. Proposition 8.4. I may be functorially extended to Q -Chu transforms such that, for every Q -functor F : E / / D , I (( Y E ) ♮ ( F,F ∗ ) / / ( Y D ) ♮ ) = F ! : P E / / P D . Hence, under the restriction to small objects, one has the commutative diagram: Q - Cat Q - Cat P / / Q - Chu Q - Cat ? ? ( Y ✷ ) ♮ ⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧⑧ Q - Chu Q - Cat I (cid:31) (cid:31) ❄❄❄❄❄❄❄❄❄❄❄ Proof.
First, for a Q -Chu transform ( F, G ) : Φ / / Ψ, one composes F ! : P E / / P D with thereflector of I Φ (cid:31) (cid:127) / / P D to define I ( F, G ) := Ψ ↓ Ψ ↑ F ! : I Φ / / I Ψ . To see the functoriality of I , we need to check I ( H, K ) I ( F, G ) = I ( HF, GK ), i.e.,Ξ ↓ Ξ ↑ H ! Ψ ↓ Ψ ↑ F ! = Ξ ↓ Ξ ↑ H ! F ! H, K ) : Ψ / / Ξ. Trivially Ξ ↓ Ξ ↑ H ! F ! ≤ Ξ ↓ Ξ ↑ H ! Ψ ↓ Ψ ↑ F ! since 1 P D ≤ Ψ ↓ Ψ ↑ by adjunction.For the reverse inequality, since Ξ ↓ Ξ ↑ is idempotent, it suffices to prove H ! Ψ ↓ Ψ ↑ ≤ Ξ ↓ Ξ ↑ H ! .Indeed, for all ψ ∈ ob P D , H ! Ψ ↓ Ψ ↑ ψ = ((Ψ ւ ψ ) ց Ψ) ◦ H ♮ ≤ (( K ♮ ◦ (Ψ ւ ψ )) ց ( K ♮ ◦ Ψ)) ◦ H ♮ = (( K ♮ ◦ (Ψ ւ ψ )) ց (Ξ ◦ H ♮ )) ◦ H ♮ (( H, K ) is a Q -Chu transform) ≤ ( K ♮ ◦ (Ψ ւ ψ )) ց (Ξ ◦ H ♮ ◦ H ♮ ) ≤ ( K ♮ ◦ (Ψ ւ ψ )) ց Ξ ( H ♮ ⊣ H ♮ )= (( K ♮ ◦ Ψ) ւ ψ ) ց Ξ (see explanation below)= ((Ξ ◦ H ♮ ) ւ ψ ) ց Ξ ((
H, K ) is a Q -Chu transform)= (Ξ ւ ( ψ ◦ H ♮ )) ց Ξ (by (b) in the proof of Theorem 6.2)= Ξ ↓ Ξ ↑ H ! ψ. Here the fourth equality from the bottom holds since one easily derives K ♮ ◦ Ψ = K ♮ ց Ψ for anyΨ, and consequently K ♮ ◦ (Ψ ւ ψ ) = K ♮ ց (Ψ ւ ψ ) = ( K ♮ ց Ψ) ւ ψ = ( K ♮ ◦ Ψ) ւ ψ. Next, for every ϕ ∈ ob P E , it is easy to see that( Y E ) ♮ ւ ϕ = P E ( ϕ, − ) ∈ ob P † E . Consequently, for all X ∈ ob E , with a repeated application of the Yoneda Lemma one obtains((( Y E ) ♮ ւ ϕ ) ց ( Y E ) ♮ ) X = ^ ψ ∈ ob P E P E ( ϕ, ψ ) ց P E ( Y E X, ψ )= P † ( P E )( P E ( Y E X, − ) , P E ( ϕ, − ))= P E ( Y E X, ϕ ) = ϕ X , and therefore I ( Y E ) ♮ = P E . Theorem 8.5.
The following assertions on a Q -category E are equivalent: (i) E is total; (ii) E is equivalent to a full reflective Q -subcategory of some presheaf Q -category; (iii) E is equivalent to the Isbell Q -category of some Q -distributor.Proof. (i) = ⇒ (ii): E is equivalent to its image under the Yoneda Q -functor Y E , which is reflectivein P E when E is total.(ii) = ⇒ (iii): We may assume that there is a full inclusion J : E (cid:31) (cid:127) / / P D with left adjoint L ,for some Q -category D . Then one defines a Q -distributor Φ : D / / ◦ E with Φ( X, ψ ) = ψ X for all X ∈ ob D , ψ ∈ ob E ⊆ ob P D . Now it is easy to verify the following calculation for all ϕ ∈ ob P D : Lϕ = ^ ψ ∈ ob P D ( Lψ ւ ϕ ) ց Lψ = ^ ψ ∈ ob P D (Φ( − , Lψ ) ւ ϕ ) ց Φ( − , Lψ )= Φ ↓ Φ ↑ ϕ. Consequently, JL = Φ ↓ Φ ↑ , from which one derives E = I Φ, as desired.(iii) = ⇒ (i): Since an Isbell Q -category is a full reflective Q -subcategory of a presheaf Q -category, which by Corollary 7.3 is total, it suffices to verify that a full reflective Q -subcategory A of a total Q -category C is total. Indeed, if L ⊣ J : A (cid:31) (cid:127) / / C , then L sup C J ! serves as a left adjointto Y A . 17 . Characterization of the Isbell Q -category of Q -distributors Theorem 8.5 shows that every total Q -category can be seen as the Isbell Q -category of some Q -distributor. Conversely, given a Q -distributor Φ, we now want to characterize its Isbell Q -category,as follows. Theorem 9.1 (Shen-Zhang [23]) . Let
Φ : E / / ◦ D be a Q -distributor. Then a Q -category C isequivalent to I Φ if, and only if, C is total and there are Q -functors F : E / / C , G : D / / C with (1) Φ = G ♮ ◦ F ♮ ; (2) F is dense (so that every object Z in C is presentable as Z ∼ = ϕ ⋆ F for some ϕ ∈ ob P E ); (3) G is codense, that is: G op is dense.Proof. The conditions are certainly necessary: given Φ and assuming C = I Φ (cid:31) (cid:127) / / P E , one defines F, G by F X = Φ ↓ Φ ↑ Y E X, GY = Φ ↓ Y †D Y for all X ∈ ob E , Y ∈ ob D . Then C ( F X, GY ) = P E (Φ ↓ Φ ↑ Y E X, Φ ↓ Y †D Y )= P † D (Φ ↑ Y E X, Y †D Y )= P E ( Y E X, Φ ↓ Y †D Y ) (Φ ↑ ⊣ Φ ↓ )= (Φ ↓ Y †D Y ) X (Yoneda Lemma)= Φ( X, Y ) , so that G ♮ ◦ F ♮ = Φ. Another straightforward calculation shows that every ϕ ∈ ob C appears asthe weighted colimit ϕ ⋆ F , so that F is dense; codensity of G follows dually.For the sufficiency of these conditions, we show that the transpose c F ♮ : C / / P E of F ♮ given by c F ♮ Z = F ♮ ( − , Z ) for all Z ∈ ob C is an equivalence of Q -categories when restricting the codomainto I Φ.First, one notices that C = F ♮ ւ F ♮ = G ♮ ց G ♮ whenever F is dense and G is codense. Indeed,by expressing each Z ∈ ob C as Z ∼ = ϕ ⋆ F for some ϕ ∈ ob P E one has C ( Z, − ) = F ♮ ւ ϕ by thedefinition of weighted colimits, and consequently C ( Z, − ) ≤ F ♮ ւ F ♮ ( − , Z ) ≤ ( F ♮ ւ F ♮ ( − , Z )) ◦ C ( Z, Z )= ( F ♮ ւ F ♮ ( − , Z )) ◦ ( F ♮ ( − , Z ) ւ ϕ ) ≤ F ♮ ւ ϕ = C ( Z, − ) , from which one derives C = F ♮ ւ F ♮ . The assertion C = G ♮ ց G ♮ follows dually.Second, the codomain of c F ♮ may be restricted to I Φ since for all Z ∈ ob P E ,Φ ↓ ( G ♮ ( Z, − )) = G ♮ ( Z, − ) ց Φ= G ♮ ( Z, − ) ց ( G ♮ ◦ F ♮ )= ( G ♮ ( Z, − ) ց G ♮ ) ◦ F ♮ = C ( − , Z ) ◦ F ♮ = F ♮ ( − , Z ) . Here the third equation holds since one easily derives Ψ ◦ F ♮ = Ψ ւ F ♮ for any Ψ, and consequently G ♮ ( Z, − ) ց ( G ♮ ◦ F ♮ ) = G ♮ ( Z, − ) ց ( G ♮ ւ F ♮ ) = ( G ♮ ( Z, − ) ց G ♮ ) ւ F ♮ = ( G ♮ ( Z, − ) ց G ♮ ) ◦ F ♮ . c F ♮ is fully faithful since for all X, X ′ ∈ ob E , I Φ( c F ♮ X, c F ♮ X ′ ) = F ♮ ( − , X ′ ) ւ F ♮ ( − , X ) = C ( X, X ′ ) . Finally, to see that c F ♮ is surjective, for each ϕ ∈ ob I Φ one forms Z = ϕ ⋆ F ∈ ob C , then G ♮ ( Z, − ) = G ♮ ◦ C ( Z, − ) = G ♮ ◦ ( F ♮ ւ ϕ ) = ( G ♮ ◦ F ♮ ) ւ ϕ = Φ ւ ϕ = Φ ↑ ϕ, and consequently c F ♮ Z = Φ ↓ ( G ♮ ( Z, − )) = Φ ↓ Φ ↑ ϕ = ϕ . Remark 9.2.
If the Q -distributor Φ in Theorem 9.1 satisfies Φ ց Φ = E , Φ ւ Φ = D , then thefunctors F : E / / I Φ, G : D / / I Φ as defined in Theorem 9.1 may be assumed to be fully faithful.Indeed, for all
X, Y ∈ ob E one then has I Φ( F X, F Y ) = P E (Φ ↓ Φ ↑ Y E X, Φ ↓ Φ ↑ Y E Y )= P † D (Φ ↑ Y E X, Φ ↑ Y E Y )= (Φ ւ Y E Y ) ց (Φ ւ Y E X )= (Φ ց Φ)(
X, Y )= E ( X, Y ) , and likewise for G . Corollary 9.3.
Let E be a full Q -subcategory of C . Then C is equivalent to the Isbell Q -category I E of (the identity Q -distributor) E if, and only if, C is total and E is both dense and codense in C .Proof. The sufficiency of the condition follows with Theorem 9.1 when one puts Φ = E , with F = G the inclusion Q -functor to C . For the necessity one involves Theorem 9.1 and Remark9.2.For a concrete category E over B , the Isbell adjunction of E (considered as the identity Q B -distributor of E ) P E P † E E ↑ / / P E P † E o o E ↓ ⊥ is described by( E ↑ ϕ ) Y = ^ X ∈ ob E E ( X, Y ) ւ ϕ X = { g ∈ B ( T, | Y | ) | ∀ X ∈ ob E , f : | X | / / T in ϕ X : g ◦ f is an E -morphism } , ( E ↓ ψ ) X = ^ Y ∈ ob E ψ Y ց E ( X, Y )= { f ∈ B ( | X | , T ) | ∀ Y ∈ ob E , g : T / / | Y | in ψ Y : g ◦ f is an E -morphism } for every structured sink ϕ in P E with codomain | ϕ | = T , and every structured source ψ in P † E with domain | ψ | = T . The full subcategory I E of P E contains the structured sinks fixed under thecorrespondence.Recall that a full subcategory E is finally dense in the concrete category C over B if every C -object is the codomain of some ( | - | )-final structured sink with domains in E . Initial density isthe dual concept. Corollary 9.4.
Let E be a full subcategory of a concrete category C over B . Then C is concretelyequivalent to I E if, and only if, C is topological over B and E is finally and initially dense in C .Proof. Final density amounts to density as a Q B -category, and initial density to codensity. Hence,Corollary 9.3 applies. 19 topological category C over B is called the MacNeille completion of its full subcategory E if E is finally and initially dense in C (see [1]). As the description of I E above shows, in that casethe category C may be built constructively from E . The smaller E may be chosen the stronger thebenefit of this fact becomes. Example 9.5. [6] For a complete lattice L (considered as a small topological category over theterminal category ) with no infinite chains, the subsets of join-irreducible elements J and of meet-irreducible elements M are respectively finally dense and initially dense in L . Let Φ : J / / ◦ M bethe order relation inherited from L between ordered sets J, M considered as categories concreteover , then the assignment x x ∩ J gives rise to the isomorphism L ∼ = I Φ. The union J ∪ M is both finally and initially dense in L , which is therefore its MacNeille completion. Example 9.6.
In the topological category
Ord over
Set of preordered sets and their monotonemaps, the full subcategory with the two-element chain as its only object is both finally andinitially dense. Indeed, for every X ∈ ob Ord , the sink
Ord ( , X ) described by all pairs ( x, y )with x ≤ y in X ) is final, and the structured source ψ with ψ = Ord ( X, ) (described by alldownsets in X ) is initial; in fact, already the source of the principal downsets ↓ x , x ∈ X is initial. Example 9.7. [11] The category
Rel of sets that comes equipped with an arbitrary relation onthem and their relation-preserving maps as morphisms is topological over
Set . The full subcate-gory with A = { , } equipped with the relation { (0 , } as its only object is finally dense; indeed,for every X ∈ ob Rel , the sink
Rel ( A, X ) describes all related pairs in X and is therefore final.An initially dense one-object full subcategory may be given by equipping the set B = { , } withthe relation { (0 , , (0 , , (1 , } ; indeed, for every X ∈ ob Rel , initiality of the source
Rel ( X, B )is easily established.While only few topological categories over
Set contain a one-object subcategory that is bothfinally and initially dense, there is a general type of topological categories over
Set that admitsa one-object initially dense subcategory. Indeed, if Q is a (small) quantale, i.e., a one-objectquantaloid, then Q becomes a Q -category whose objects are the elements of Q , and whose hom-arrows are given by Q ( u, v ) = v ւ u ( u, v ∈ Q ) . Proposition 9.8.
For a quantale Q , the functor ob : Q - Cat / / Set is topological, and the fullsubcategory with Q as its only object is initially dense in Q - Cat .Proof.
For the topologicity assertion, see [14, Theorem III.3.1.3], and for the fact that {Q} isinitially dense in Q - Cat , see [14, Exercise III.1.H]. Indeed, given a small Q -category E , initialityof the source E ( X, − ) : E / / Q ( X ∈ ob E )is easily verified. Remark 9.9.
As has been shown in [22], the first assertion of Proposition 9.8 easily generalizesfrom quantales to quantaloids: Q - Cat is topological over
Set / ob Q for any small quantaloid Q .For Q = the two-element chain (with ◦ given by meet), Proposition 9.8 reproduces theinitiality assertion for in Ord = - Cat . For Q = ([0 , ∞ ] , ≥ ) (with ◦ given by +), Q - Cat = Met is Lawvere’s category of generalized metric spaces (
X, d ) (with d : X × X / / [0 , ∞ ] satisfying d ( x, x ) = 0 and d ( x, z ) ≤ d ( x, y ) + d ( y, z ) for all x, y, z ∈ X ) and their non-expanding maps f : ( X, d ) / / ( Y, e ) (with e ( f ( x ) , f ( y )) ≤ d ( x, y ) for all x, y ∈ X ). Example 9.10. ([0 , ∞ ] , h ) with h ( r, s ) = s − r, if r ≤ s < ∞ , , if s ≤ r, ∞ , if r < s = ∞ is initially dense in Met but not finally dense. 20
0. Total Q -categories as injective objects It is well-known (see [27]) that small total Q -categories are characterized as the injective objectsin Q - Cat . For a large Q -category E , the standard proof which employs P E as a test object, hasto be modified as P E may be illegitimately large. We therefore give a modified proof that is validalso in the large case. Theorem 10.1 (Stubbe [27]) . A Q -category is total if, and only if, it is injective in Q - CAT w.r.t. fully faithful Q -functors.Proof. For E total and Q -functors F : C / / E , G : C / / D with G fully faithful we must find H : D / / E with HG ∼ = F . With c G ♮ denoting the transpose of G ♮ : C / / ◦ D , we may define H = ( D c G ♮ / / P C F ! / / P E sup / / E ) . One then has, for all X ∈ ob E , HGX = sup F ! ( G ♮ ( − , GX ))= sup F ! ( D ( G − , GX ))= sup F ! Y C X ( G fully faithful)= sup Y E F X ( Y : 1 / / P natural) ∼ = F X.
Conversely, for a Q -category E and ϕ ∈ ob P E , in order to find the supremum of ϕ , one mayexploit its injectivity in Q - CAT on the fully faithful Q -functor Y E : E / / D , where D is the Q -subcategory of the Q -(meta)category P E with ob D = { Y E X | X ∈ ob E} ∪ { ϕ } . Thus oneobtains a Q -functor H : D / / E with H Y E = 1 E . Note that for all X ∈ ob E , E ( Hϕ, X ) = P E ( Y E Hϕ, Y E X ) (Yoneda Lemma)= Y E X ւ E ( − , Hϕ )= Y E X ւ E ( H Y E − , Hϕ ) ( H Y E = 1 E ) ≤ Y E X ւ D ( Y E − , ϕ )= Y E X ւ ϕ (Yoneda Lemma)= D ( ϕ, Y E X ) ≤ E ( Hϕ, H Y E X )= E ( Hϕ, X ) . ( H Y E = 1 E )Therefore E ( Hϕ, X ) = Y E X ւ ϕ = P E ( ϕ, Y E X ) for all X ∈ ob E and, consequently, Hϕ is thesupremum of ϕ .Exploiting Theorem 10.1 for Q = Q B , where B is an ordinary category, one reproduces aclassical result of categorical topology: Corollary 10.2 (Br¨ummer-Hoffmann [5]) . For a topological category E over B and concrete func-tors F : C / / E , G : C / / D over B with G fully faithful, there is a concrete functor H : D / / E with HG ∼ = F concretely isomorphic. Conversely, this property characterizes topologicity of E over B . Strictly speaking, “injective” should read “ quasi-injective ”, since, in the proof below, generally we obtain H only with HG ∼ = F , not HG = F . eferences [1] J. Ad´amek, H. Herrlich, and G. E. Strecker. Abstract and Concrete Categories: The Joy ofCats . Wiley, New York, 1990.[2] M. Barr. ∗ -Autonomous categories and linear logic. Mathematical Structures in ComputerScience , 1:159–178, 1991.[3] J. B´enabou. Les distributeurs. Universit´e Catholique de Louvain, Institute de Mat´ematiquePure et Appliqu´ee, Rapport no. 33, 1973.[4] G. C. L. Br¨ummer.
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