The enriched Vietoris monad on representable spaces

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Abstract

Employing a formal analogy between ordered sets and topological spaces, over the past years we have investigated a notion of cocompleteness for topological, approach and other kind of spaces. In this new context, the down-set monad becomes the filter monad, cocomplete ordered set translates to continuous lattice, distributivity means disconnectedness, and so on. Curiously, the dual(?) notion of completeness does not behave as the mirror image of the one of cocompleteness; and in this paper we have a closer look at complete spaces. In particular, we construct the “up-set monad” on representable spaces (in the sense of L. Nachbin for topological spaces, respectively C. Hermida for multicategories); we show that this monad is of Kock–Zöberlein type; we introduce and study a notion of weighted limit similar to the classical notion for enriched categories; and we describe the Kleisli category of our “up-set monad”. We emphasise that these generic categorical notions and results can be indeed connected to more “classical” topology: for topological spaces, the “up-set monad” becomes the lower Vietoris monad, and the statement “X is totally cocomplete if and only if Xop is totally complete” specialises to O. Wyler's characterisation of the algebras of the Vietoris monad on compact Hausdorff spaces as precisely the continuous lattices.

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 typeXop×X2, and from that one obtains the two Yoneda embeddingsX2Xop=:PXandX(2X)op=:VX. Furthermore, both constructions are part of monads P and V on Ord (the category of ordered sets and monotone maps) with Eilenberg–Moore categories OrdPSup (the category of complete lattices and sup-preserving maps) and OrdVInf (the category of complete lattices and inf-preserving maps) respectively. One has full embeddingsOrdPOrdPandOrdVOrdV from the Kleisli categories into the Eilenberg–Moore categories, and from that one obtains equivalences (see [50], [51])kar(OrdP)CCDsupandkar(OrdV)CCDinfop 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 (opccd)-lattices and inf-preserving maps on the other. Here a complete lattice L is (opccd) whenever Lop is completely distributive; in fact, L is (opccd) if and only if L is (ccd). These equivalences restrict toOrdPTalsupandOrdVTalinfop, where “Tal” stands for totally algebraic lattice and a lattice is in Talinfop whenever Lop is in Talsup. Finally, having both sides restricted to adjoint morphisms leads to the equivalencesTalOrdopTalop between the dual category of Ord and the category Tal (respectively Talop) 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(UX)op×X2 (where 2 is the Sierpiński space and (UX)op is explained in Section 2). Moreover, the space (UX)op turns out to be exponentiable, therefore we obtain the Yoneda embeddingyX:X2(UX)op=:PX. 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

  • In Section 1 we describe our general framework, namely that of a topological theory T=(T,V,ξ) (see [26]) consisting of a monad T=(T,e,m) on Set, a quantale V and a T-algebra structure ξ:TVV on V. The associated notion of T-category embodies several types of spaces such as topological, metric or approach spaces, and together with T-functors and T-distributors defines the categories T-Cat and T-Dist 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 T-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).

  • Section 2 is devoted to the important notion of representable T-categories (Definition 2.5) defined as precisely the pseudo-algebras for a natural lifting of the Set-monad T to a monad of Kock–Zöberlein type on T-Cat. We also introduce the concept of a dualisable T-graph (Definition 2.10). Our interest in representable T-categories derives from the fact that these are precisely those T-categories for which the associated dual T-graph is a T-category (Definition 2.12 and Proposition 2.15). For topological T0-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.

  • In Section 3 we recall the principal facts about weighted colimits and cocomplete T-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 T-category admitting all weighted colimits, and say that a T-category is “cocomplete” whenever it has all those weighted colimits where the codomain of the weight is G.

  • From Section 4 on we assume that our monad T=(T,e,m) satisfies T1=1. In this section we show that the exponential VX is always a dualisable T-graph and its dual (VX)op is always a T-category. In the topological case, (2X)op 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 X(VX)op=:VX leads to a monad V=(V,h,w) of Kock–Zöberlein type on both T-Cat and T-ReprCat (the category of representable T-categories and pseudo-homomorphisms), see Theorem 4.20.

  • In Section 5 we analyse the notion of weighted limit in T-categories.

  • Section 6 lifts the classical adjunction between “up-sets” and “down-sets” (see [63, Section 5]) into the realm of T-categories.

  • In Section 7 we introduce totally complete T-categories and show that they are precisely the duals of totally cocomplete T-categories (Theorem 7.4 and Examples 7.6).

  • Finally, in Section 8 we give a characterisation of the morphisms of the Kleisli category T-ReprCatV of V. We also observe how the notion of an Esakia space arises naturally in this context via splitting idempotents of the full subcategory of T-ReprCatV defined by all T-algebras. We find it worthwhile to mention that this implies in particular that the category coHeyt, of co-Heyting algebras and finite suprema preserving maps is the idempotent split completion of the category Bool, of Boolean algebras and finite suprema preserving maps (Example 8.11).

Section snippets

The setting

In this paper we deal with T-categories, T-functors and T-distributors, for a topological theory T. Below we recall some of the main facts and refer to [26], [27] and [10] for details.

Definition 1.1

A topological theory T=(T,V,ξ) consists of:

  • (1)

    a monad T=(T,e,m) on Set (with multiplication m and unit e),

  • (2)

    a commutative and unital quantale V=(V,,k),

  • (3)

    a function ξ:TVV,

such that
  • (a)

    T preserves weak pullbacks and each naturality square of m is a weak pullback,

  • (b)

    the pair (V,ξ) is an Eilenberg–Moore algebra for T and the

Representable T-categories and dualisation

In [9] we introduced a notion of dual T-category as a crucial step towards the Yoneda lemma and related results. The basic idea is to associate to a T-category X a V-category MX which still contains all information about the T-categorical structure of X, and then use the usual dualisation of V-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 T-categories

By an appropriate translation from the V to the T-case one can transport the notions of weighted colimit (see [16], [36]) and cocompleteness into the realm of T-categories, as we recall in this section briefly from [27] and [10]. A weighted colimit diagram in a T-category X is given by a T-functor d:DX and a T-distributor φ:DG (where G=(1,e1)). A colimit of such weighted diagram is an element xX which represents dφ, that is, x=dφ. 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, T=(T,V,ξ) denotes a (strict) topological theory where, moreover, T1=1.

Under these conditions, the T-category V=(V,homξ)P1 is totally cocomplete. Furthermore, since e1:1T1 is a bijection, we can identify a V-relation φ:1X with the T-relation φe1:1X. If, moreover, X=(X,a) is a T-category, then φe1 is a T-distributor of type GX if and only ifaTξφe1φ. Note that aTξφe1φ holds for every V-relation φ:1X.

Lemma 4.1

For every T-category X=(X,a

Complete T-categories

Similar to what was done for colimits, we introduce now a notion of weighted limit for T-categories following closely the V-categorical case. A weighted limit diagram (h,φ) in a T-category X is given by a T-functor h:AX and a T-distributor φ:GA, and x0X is a limit of this diagram, written as x0lim(h,φ), if x0 represents φh in the sense that x0=φh. We hasten to remark that we cannot consider an arbitrary T-distributor φ:BA above since the lifting φh might not exist. A T-functor f:XY

Isbell conjugation adjunction

For every T-category X=(X,a), there is an adjunction in V-Cat whereφ(x)=φ1X(x)=xXhom(φ(x),a(x,x)) for all φ:GX and xTX, andψ+(x)=1Xψ(x)=xTXhom(ψ(x),a(x,x)) for all ψ:XG and xX. Following [63], we refer to this adjunction as an Isbell conjugation adjunction. Note that(x)=xand(x)+=x, for all xX. We will see that () is even a T-functor of type ():VXPX, but ()+ fails in general to be a T-functor.

Proposition 6.1

For every T-category X=(X,a), the V-functor () is actually a T-functor ():VXPX

Totally complete T-categories

At the beginning of Section 5 we pointed already out that the notion of complete T-category cannot be strengthened to “totally complete” exactly the same way as it was done for cocompleteness, namely by allowing all T-distributors ψ:BA 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 T-category X=(X,a) is called totally complete if hX:XVX has a right adjoint InfX:VXX in T-ReprCat.

The Kleisli category of the Vietoris monad

Every T-functor r:XVY gives rise to a V-matrix r:XY where r(x,y)=r(x)(y). In the sequel we are interested in the case where X=(X,a) and Y=(Y,b) are representable T-categories and r:XVY is a pseudo-homomorphism.

Proposition 8.1

Let X=(X,a) and Y=(Y,b) be representable T-categories. Then rr defines a bijection between T-ReprCat(X,VY) and the subset of V-Dist(X0,Y0) consisting of all those V-distributors ψ:X0Y0 making the diagram of V-distributors commutative.

Proof

Let α:TXX and β:TYY 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.

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