The enriched Vietoris monad on representable spaces☆
Introduction
In this paper we continue the work presented in [27] on “injective spaces via adjunction” whose fundamental aspect can be described by the slogan topological spaces are categories, and therefore can be studied using notions and techniques from (enriched) Category Theory. The use of the word “continue” here is slightly misleading as we do not follow directly the path of [27] but rather develop “the second aspect” of this theory. To explain this better, recall that an order relation on a set X defines a monotone map of type and from that one obtains the two Yoneda embeddings Furthermore, both constructions are part of monads and on (the category of ordered sets and monotone maps) with Eilenberg–Moore categories (the category of complete lattices and sup-preserving maps) and (the category of complete lattices and inf-preserving maps) respectively. One has full embeddings from the Kleisli categories into the Eilenberg–Moore categories, and from that one obtains equivalences (see [50], [51]) between the idempotent split completion of the Kleisli categories on one side and the categories of completely distributive complete lattices ((ccd)-lattices for short) and sup-preserving maps respectively -lattices and inf-preserving maps on the other. Here a complete lattice L is whenever is completely distributive; in fact, L is if and only if L is (ccd). These equivalences restrict to where “Tal” stands for totally algebraic lattice and a lattice is in whenever is in . Finally, having both sides restricted to adjoint morphisms leads to the equivalences between the dual category of and the category (respectively ) with morphisms all sup- and inf-preserving maps.
In [27], [28] and [10] we followed the path on the left described above, but now with geometric objects like topological or approach spaces in lieu of ordered sets (the latter representing “metric” topological spaces, see [42]). To illustrate this analogy, note that the ultrafilter convergence of a topological space defines a continuous map (where is the Sierpiński space and is explained in Section 2). Moreover, the space turns out to be exponentiable, therefore we obtain the Yoneda embedding The “story of cocompleteness” can now be told almost as for ordered sets, and we refer to the above-mentioned papers for detailed information. However, in contrast to the ordered case, the subsequent development of the right side cannot be seen as the dual image of the left side; and it is the aim of this work to explore this path.
The paper is organised as follows
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In Section 1 we describe our general framework, namely that of a topological theory (see [26]) consisting of a monad on , a quantale and a -algebra structure on . The associated notion of -category embodies several types of spaces such as topological, metric or approach spaces, and together with -functors and -distributors defines the categories and respectively. We recall succinctly the main constructions and results, in particular that core-compactness implies exponentiability with respect to a naturally defined tensor product of -categories. In this context, for topological spaces we give a variation of Alexander's Subbase Lemma for core-compactness using a simple convergence-theoretic argument (Example 1.9).
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Section 2 is devoted to the important notion of representable -categories (Definition 2.5) defined as precisely the pseudo-algebras for a natural lifting of the -monad to a monad of Kock–Zöberlein type on . We also introduce the concept of a dualisable -graph (Definition 2.10). Our interest in representable -categories derives from the fact that these are precisely those -categories for which the associated dual -graph is a -category (Definition 2.12 and Proposition 2.15). For topological -spaces, the concept of representability specialises to the classical notion of a stably compact space which is closely related to L. Nachbin's ordered compact Hausdorff spaces.
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In Section 3 we recall the principal facts about weighted colimits and cocomplete -categories obtained in Hofmann [27] and Clementino and Hofmann [10]. We stress that, unlike the classical case of enriched categories, here it is necessary to consider weights with arbitrary codomain, not just the one-element category G. Compared to previous work we change notation and use the designation “totally cocomplete” for a -category admitting all weighted colimits, and say that a -category is “cocomplete” whenever it has all those weighted colimits where the codomain of the weight is G.
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From Section 4 on we assume that our monad satisfies . In this section we show that the exponential is always a dualisable -graph and its dual is always a -category. In the topological case, turns out to be the lower Vietoris topological space; and we point out how this can be used to deduce the classical characterisation of exponentiable topological spaces as precisely the core-compact ones (Example 4.3). The main result of this section states that the construction leads to a monad of Kock–Zöberlein type on both and (the category of representable -categories and pseudo-homomorphisms), see Theorem 4.20.
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In Section 5 we analyse the notion of weighted limit in -categories.
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Section 6 lifts the classical adjunction between “up-sets” and “down-sets” (see [63, Section 5]) into the realm of -categories.
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In Section 7 we introduce totally complete -categories and show that they are precisely the duals of totally cocomplete -categories (Theorem 7.4 and Examples 7.6).
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Finally, in Section 8 we give a characterisation of the morphisms of the Kleisli category of . We also observe how the notion of an Esakia space arises naturally in this context via splitting idempotents of the full subcategory of defined by all -algebras. We find it worthwhile to mention that this implies in particular that the category of co-Heyting algebras and finite suprema preserving maps is the idempotent split completion of the category of Boolean algebras and finite suprema preserving maps (Example 8.11).
Section snippets
The setting
In this paper we deal with -categories, -functors and -distributors, for a topological theory . Below we recall some of the main facts and refer to [26], [27] and [10] for details.
Definition 1.1 A topological theory consists of: a monad on (with multiplication m and unit e), a commutative and unital quantale , a function , T preserves weak pullbacks and each naturality square of m is a weak pullback, the pair is an Eilenberg–Moore algebra for and the
such that
Representable -categories and dualisation
In [9] we introduced a notion of dual -category as a crucial step towards the Yoneda lemma and related results. The basic idea is to associate to a -category X a -category MX which still contains all information about the -categorical structure of X, and then use the usual dualisation of -categories. Later, in [21], [28], we noted already that this construction is closely related to Nachbin's ordered compact Hausdorff spaces [46] as presented in [60]. In this section we continue this path
Cocomplete -categories
By an appropriate translation from the to the -case one can transport the notions of weighted colimit (see [16], [36]) and cocompleteness into the realm of -categories, as we recall in this section briefly from [27] and [10]. A weighted colimit diagram in a -category X is given by a -functor and a -distributor (where ). A colimit of such weighted diagram is an element which represents , that is, . If such x exists, it is unique up to equivalence, and one
The Vietoris monad
General assumption. From now on until the end of this paper, denotes a (strict) topological theory where, moreover, .
Under these conditions, the -category is totally cocomplete. Furthermore, since is a bijection, we can identify a -relation with the -relation . If, moreover, is a -category, then is a -distributor of type if and only if Note that holds for every -relation .
Lemma 4.1 For every -category
Complete -categories
Similar to what was done for colimits, we introduce now a notion of weighted limit for -categories following closely the -categorical case. A weighted limit diagram in a -category X is given by a -functor and a -distributor , and is a limit of this diagram, written as , if represents in the sense that . We hasten to remark that we cannot consider an arbitrary -distributor above since the lifting might not exist. A -functor
Isbell conjugation adjunction
For every -category , there is an adjunction in where for all and , and for all and . Following [63], we refer to this adjunction as an Isbell conjugation adjunction. Note that for all . We will see that is even a -functor of type , but fails in general to be a -functor.
Proposition 6.1 For every -category , the -functor is actually a -functor
Totally complete -categories
At the beginning of Section 5 we pointed already out that the notion of complete -category cannot be strengthened to “totally complete” exactly the same way as it was done for cocompleteness, namely by allowing all -distributors as limit weights. Nevertheless, in this section we introduce a notion of total completeness which turns out to be the dual of total cocompleteness.
Definition 7.1 A representable -category is called totally complete if has a right adjoint in .
The Kleisli category of the Vietoris monad
Every -functor gives rise to a -matrix where . In the sequel we are interested in the case where and are representable -categories and is a pseudo-homomorphism. Proposition 8.1 Let and be representable -categories. Then defines a bijection between and the subset of consisting of all those -distributors making the diagram of -distributors commutative. Proof Let and be pseudo-algebra
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Partial financial assistance by Portuguese funds through CIDMA (Center for Research and Development in Mathematics and Applications), and the Portuguese Foundation for Science and Technology (“FCT – Fundação para a Ciência e a Tecnologia”), within the project PEst-OE/MAT/UI4106/2014, and by the project NASONI under the contract PTDC/EEI-CTP/2341/2012 is gratefully acknowledged.