Search

Compact metric spaces form an important class of metric spaces, but the category that they define lacks many important properties such as completeness and cocompleteness. In recent studies of "metric domain theory" and Stone-type dualities, the more general notion of a (separated) metric compact Hausdorff space emerged as a metric counterpart of Nachbin's compact ordered spaces. Roughly speaking, a metric compact Hausdorff space is a metric space equipped with a \emph{compatible} compact Hausdorff topology (which does not need to be the induced topology). These spaces maintain many important features of compact metric spaces, and, notably, the resulting category is much better behaved. Moreover, one can use inspiration from the theory of Nachbin's compact ordered spaces to solve problems for metric structures. In this paper we continue this line of research: in the category of separated metric compact Hausdorff spaces we characterise the regular monomorphisms as the embeddings and the epimorphisms as the surjective morphisms. Moreover, we show that epimorphisms out of an object $X$ can be encoded internally on $X$ by their kernel metrics, which are characterised as the continuous metrics below the metric on $X$; this gives a convenient way to represent quotient objects. Finally, as the main result, we prove that its dual category has an algebraic flavour: it is Barr-exact. While we show that it cannot be a variety of finitary algebras, it remains open whether it is an infinitary variety.

Building on the notion of normed category as suggested by Lawvere, we introduce notions of Cauchy convergence and cocompleteness which differ from proposals in previous works. Key to our approach is to treat them consequentially as categories enriched in the monoidal-closed category of normed sets. Our notions largely lead to the anticipated outcomes when considering individual metric spaces as small normed categories, but they can be challenging when considering some large categories, like those of semi-normed or normed vector spaces and all linear maps, or of generalized metric spaces and all mappings. These are the key example categories discussed in detail in this paper. Working with a general commutative quantale V as a value recipient for norms, rather than only with Lawvere's quantale of the extended real half-line, we observe that the categorically atypical structure gap between objects and morphisms in the example categories is already present in the underlying normed category of the enriching category of V-normed sets. To show that this normed category and, in fact, all presheaf categories over it, are Cauchy cocomplete, we assume the quantale V to satisfy a couple of light alternative extra properties. Of utmost importance to the general theory is the fact that our notion of normed colimit is subsumed by the notion of weighted colimit of enriched category theory. With this theory we are able to prove that all V-normed categories have correct-size Cauchy cocompletions. We also prove a Banach Fixed Point Theorem for contractive endofunctors of Cauchy cocomplete normed categories., Comment: 50 pages

It is shown that the duals of several categories of topological flavour, like the categories of ordered sets, generalised metric spaces, probabilistic metric spaces, topological spaces, approach spaces, are quasivarieties, presenting a common proof for all such results.

The classical Hennessy-Milner theorem is an important tool in the analysis of concurrent processes; it guarantees that any two non-bisimilar states in finitely branching labelled transition systems can be distinguished by a modal formula. Numerous variants of this theorem have since been established for a wide range of logics and system types, including quantitative versions where lower bounds on behavioural distance (e.g.~in weighted, metric, or probabilistic transition systems) are witnessed by quantitative modal formulas. Both the qualitative and the quantitative versions have been accommodated within the framework of coalgebraic logic, with distances taking values in quantales, subject to certain restrictions, such as being so-called value quantales. While previous quantitative coalgebraic Hennessy-Milner theorems apply only to liftings of set functors to (pseudo-)metric spaces, in the present work we provide a quantitative coalgebraic Hennessy-Milner theorem that applies more widely to functors native to metric spaces; notably, we thus cover, for the first time, the well-known Hennessy-Milner theorem for continuous probabilistic transition systems, where transitions are given by Borel measures on metric spaces, as an instance. In the process, we also relax the restrictions imposed on the quantale, and additionally parametrize the technical account over notions of closure and, hence, density, providing associated variants of the Stone-Weierstrass theorem; this allows us to cover, for instance, behavioural ultrametrics.

Behavioural distances measure the deviation between states in quantitative systems, such as probabilistic or weighted systems. There is growing interest in generic approaches to behavioural distances. In particular, coalgebraic methods capture variations in the system type (nondeterministic, probabilistic, game-based etc.), and the notion of quantale abstracts over the actual values distances take, thus covering, e.g., two-valued equivalences, (pseudo-)metrics, and probabilistic (pseudo-)metrics. Coalgebraic behavioural distances have been based either on liftings of SET-functors to categories of metric spaces, or on lax extensions of SET-functors to categories of quantitative relations. Every lax extension induces a functor lifting but not every lifting comes from a lax extension. It was shown recently that every lax extension is Kantorovich, i.e. induced by a suitable choice of monotone predicate liftings, implying via a quantitative coalgebraic Hennessy-Milner theorem that behavioural distances induced by lax extensions can be characterized by quantitative modal logics. Here, we essentially show the same in the more general setting of behavioural distances induced by functor liftings. In particular, we show that every functor lifting, and indeed every functor on (quantale-valued) metric spaces, that preserves isometries is Kantorovich, so that the induced behavioural distance (on systems of suitably restricted branching degree) can be characterized by a quantitative modal logic.

Lax extensions of set functors play a key role in various areas including topology, concurrent systems, and modal logic, while predicate liftings provide a generic semantics of modal operators. We take a fresh look at the connection between lax extensions and predicate liftings from the point of view of quantale-enriched relations. Using this perspective, we show in particular that various fundamental concepts and results arise naturally and their proofs become very elementary. Ultimately, we prove that every lax extension is induced by a class of predicate liftings; we discuss several implications of this result.

The term Stone-type duality often refers to a dual equivalence between a category of lattices or other partially ordered structures on one side and a category of topological structures on the other. This paper is part of a larger endeavour that aims to extend a web of Stone-type dualities from ordered to metric structures and, more generally, to quantale-enriched categories. In particular, we improve our previous work and show how certain duality results for categories of [0,1]-enriched Priestley spaces and [0,1]-enriched relations can be restricted to functions. In a broader context, we investigate the category of quantale-enriched Priestley spaces and continuous functors, with emphasis on those properties which identify the algebraic nature of the dual of this category.

As composites of constant, (co)product, identity, and powerset functors, Kripke polynomial functors form a relevant class of $\mathsf{Set}$-functors in the theory of coalgebras. The main goal of this paper is to expand the theory of limits in categories of coalgebras of Kripke polynomial functors to the context of quantale-enriched categories. To assume the role of the powerset functor we consider "powerset-like" functors based on the Hausdorff $\mathsf{V}$-category structure. As a starting point, we show that for a lifting of a $\mathsf{SET}$-functor to a topological category $\mathsf{X}$ over $\mathsf{Set}$ that commutes with the forgetful functor, the corresponding category of coalgebras over $\mathsf{X}$ is topological over the category of coalgebras over $\mathsf{Set}$ and, therefore, it is "as complete" but cannot be "more complete". Secondly, based on a Cantor-like argument, we observe that Hausdorff functors on categories of quantale-enriched categories do not admit a terminal coalgebra. Finally, in order to overcome these "negative" results, we combine quantale-enriched categories and topology \emph{\`a la} Nachbin. Besides studying some basic properties of these categories, we investigate "powerset-like" functors which simultaneously encode the classical Hausdorff metric and Vietoris topology and show that the corresponding categories of coalgebras of "Kripke polynomial" functors are (co)complete.

Using generalized enriched categories, in this paper we show that Rosick\'{y}'s proof of cartesian closedness of the exact completion of the category of topological spaces can be extended to a wide range of topological categories over $\mathsf{Set}$, like metric spaces, approach spaces, ultrametric spaces, probabilistic metric spaces, and bitopological spaces. In order to do so we prove a sufficient criterion for exponentiability of $(\mathbb{T},V)$-categories and show that, under suitable conditions, every $(\mathbb{T},V)$-injective category is exponentiable in $(\mathbb{T},V)\text{-}\mathsf{Cat}$.

It is known since the late 1960's that the dual of the category of compact Hausdorff spaces and continuous maps is a variety -- not finitary, but bounded by $\aleph_1$. In this note we show that the dual of the category of partially ordered compact spaces and monotone continuous maps is a $\aleph_1$-ary quasivariety, and describe partially its algebraic theory. Based on this description, we extend these results to categories of Vietoris coalgebras and homomorphisms. We also characterise the $\aleph_1$-copresentable partially ordered compact spaces.

Generalising Nachbin's theory of "topology and order", in this paper we continue the study of quantale-enriched categories equipped with a compact Hausdorff topology. We compare these $\mathcal{V}$-categorical compact Hausdorff spaces with ultrafilter-quantale-enriched categories, and show that the presence of a compact Hausdorff topology guarantees Cauchy completeness and (suitably defined) codirected completeness of the underlying quantale enriched category.

In this paper we investigate important categories lying strictly between the Kleisli category and the Eilenberg-Moore category, for a Kock-Z\"oberlein monad on an order-enriched category. Firstly, we give a characterisation of free algebras in the spirit of domain theory. Secondly, we study the existence of weighted (co)limits, both on the abstract level and for specific categories of domain theory like the category of algebraic lattices. Finally, we apply these results to give a description of the idempotent split completion of the Kleisli category of the filter monad on the category of topological spaces., Comment: small corrections

Motivated by the need to reason about hybrid systems, we study limits in categories of coalgebras whose underlying functor is a Vietoris polynomial one - intuitively, the topological analogue of a Kripke polynomial functor. Among other results, we prove that every Vietoris polynomial functor admits a final coalgebra if it respects certain conditions concerning separation axioms and compactness. When the functor is restricted to some of the categories induced by these conditions the resulting categories of coalgebras are even complete. As a practical application, we use these developments in the specification and analysis of non-deterministic hybrid systems, in particular to obtain suitable notions of stability, and behaviour.

For any small quantaloid $\Q$, there is a new quantaloid $\D(\Q)$ of diagonals in $\Q$. If $\Q$ is divisible then so is $\D(\Q)$ (and vice versa), and then it is particularly interesting to compare categories enriched in $\Q$ with categories enriched in $\D(\Q)$. Taking Lawvere's quantale of extended positive real numbers as base quantale, $\Q$-categories are generalised metric spaces, and $\D(\Q)$-categories are generalised partial metric spaces, i.e.\ metric spaces in which self-distance need not be zero and with a suitably modified triangular inequality. We show how every small quantaloid-enriched category has a canonical closure operator on its set of objects: this makes for a functor from quantaloid-enriched categories to closure spaces. Under mild necessary-and-sufficient conditions on the base quantaloid, this functor lands in the category of topological spaces; and an involutive quantaloid is Cauchy-bilateral (a property discovered earlier in the context of distributive laws) if and only if the closure on any enriched category is identical to the closure on its symmetrisation. As this now applies to metric spaces and partial metric spaces alike, we demonstrate how these general categorical constructions produce the "correct" definitions of convergence and Cauchyness of sequences in generalised partial metric spaces. Finally we describe the Cauchy-completion, the Hausdorff contruction and exponentiability of a partial metric space, again by application of general quantaloid-enriched category theory., Comment: Apart from some minor corrections, this second version contains a revised section on Cauchy sequences in a partial metric space

A common feature of many duality results is that the involved equivalence functors are liftings of hom-functors into the two-element space resp. lattice. Due to this fact, we can only expect dualities for categories cogenerated by the two-element set with an appropriate structure. A prime example of such a situation is Stone's duality theorem for Boolean algebras and Boolean spaces,the latter being precisely those compact Hausdorff spaces which are cogenerated by the two-element discrete space. In this paper we aim for a systematic way of extending this duality theorem to categories including all compact Hausdorff spaces. To achieve this goal, we combine duality theory and quantale-enriched category theory. Our main idea is that, when passing from the two-element discrete space to a cogenerator of the category of compact Hausdorff spaces, all other involved structures should be substituted by corresponding enriched versions. Accordingly, we work with the unit interval $[0,1]$ and present duality theory for ordered and metric compact Hausdorff spaces and (suitably defined) finitely cocomplete categories enriched in $[0,1]$.

In this work, we describe an adjunction between the comma category of SET-based monads under the V-powerset monad and the category of associative lax extensions of SET-based monads to the category of V-relations. In the process, we give a general construction of the Kleisli extension of a monad to the category of V-relations.

The original purpose of component-based development was to provide techniques to master complex software, through composition, reuse and parametrisation. However, such systems are rapidly moving towards a level in which software becomes prevalently intertwined with (continuous) physical processes. A possible way to accommodate the latter in component calculi relies on a suitable encoding of continuous behaviour as (yet another) computational effect. This paper introduces such an encoding through a monad which, in the compositional development of hybrid systems, may play a role similar to the one played by the 1+, powerset, and distribution monads in the characterisation of partial, non deterministic and probabilistic components, respectively. This monad and its Kleisli category provide a setting in which the effects of continuity over (different forms of) composition can be suitably studied., Comment: Journal of Logical and Algebraic Methods in Programming, 2016

Working in the framework of $(T, V)$-categories, for a symmetric monoidal closed category $V$ and a (not necessarily cartesian) monad $T$, we present a common account to the study of ordered compact Hausdorff spaces and stably compact spaces on one side and monoidal categories and representable multicategories on the other one. In this setting we introduce the notion of dual for $(T, V)$-categories.

Under Stone/Priestley duality for distributive lattices, Esakia spaces correspond to Heyting algebras which leads to the well-known dual equivalence between the category of Esakia spaces and morphisms on one side and the category of Heyting algebras and Heyting morphisms on the other. Based on the technique of idempotent split completion, we give a simple proof of a more general result involving certain relations rather then functions as morphisms. We also extend the notion of Esakia space to all stably locally compact spaces and show that these spaces define the idempotent split completion of compact Hausdorff spaces. Finally, we exhibit connections with split algebras for related monads.

In this note we present a characterisation of exponentiable approach spaces in terms of ultrafilter convergence.

In this paper we show how the theory of monads can be used to deduce in a uniform manner several duality theorems involving categories of relations on one side and categories of algebras with homomorphisms preserving only some operations on the other. Furthermore, we investigate the monoidal structure induced by Cartesian product on the relational side and show that in some cases the corresponding operation on the algebraic side represents bimorphisms.

We investigate the time an electronic excitation travels in a supermolecular setup using a measurement process in an open quantum-system framework. The approach is based on the stochastic Schr\"odinger equation and uses a Hamiltonian from time-dependent density functional theory (TDDFT). It treats electronic-structure properties and intermolecular coupling on the level of TDDFT, while it opens a route to the description of dissipation and relaxation via a bath operator that couples to the dipole moment of the density. Within our study, we find that in supermolecular setups small deviations of the electronic structure from the perfectly resonant case have only minor influence on the pathways of excitation-energy transfer, thus lead to similar transfer times. Yet, sizable defects cause notable slowdown of the energy spread.

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-Z\"oberlein 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 emphasize that these generic categorical notions and results can be indeed connected to more "classical" topology: for topological spaces, the "up-set monad" becomes the upper Vietoris monad, and the statement "$X$ is totally cocomplete if and only if $X^\mathrm{op}$ is totally complete" specialises to O. Wyler's characterisation of the algebras of the Vietoris monad on compact Hausdorff spaces., Comment: One error in Example 1.9 is corrected; Section 4 works now without the assuming core-compactness

In this paper we present methods of transition from one perspective on logic to others, and apply this in particular to obtain a coalgebraic presentation of logic. The central ingredient in this process is to view consequence relations as morphisms in a category.

Lax monoidal powerset-enriched monads yield a monoidal structure on the category of monoids in the Kleisli category of a monad. Exponentiable objects in this category are identified as those Kleisli monoids with algebraic structure. This result generalizes the classical identification of exponentiable topological spaces as those whose lattice of open subsets forms a continuous lattice., Comment: v2: minor typos corrected

In this paper we investigate Cauchy completeness and exponentiablity for quantale enriched categories, paying particular attention to probabilistic metric spaces.

In analogy to the situation for continuous lattices which were introduced by Dana Scott as precisely the injective T$_0$ spaces via the (nowadays called) Scott topology, we study those metric spaces which correspond to injective T$_0$ approach spaces and characterise them as precisely the continuous lattices equipped with an unitary and associative $[0,\infty]$-action. This result is achieved by a thorough analysis of the notion of cocompleteness for approach spaces.

In this paper we consider topological spaces as generalised orders and characterise those spaces which satisfy a (suitably defined) topological distributive law. Furthermore, we show that the category of these spaces is dually equivalent to a certain category of frames by simply observing that both sides represent the idempotents split completion of the same category.

We describe a duality for quantale-enriched categories that extends the Lawson duality for continuous dcpos: for any saturated class J of modules that commute with certain weighted limits, and under an appropriate choice of morphisms, the category of J-cocomplete and J-continuous quantale-enriched categories is self-dual., Comment: 16 pages

The main source of inspiration for the present paper is the work of R. Rosebrugh and R.J. Wood on constructive complete distributive lattices where the authors employ elegantly the concepts of adjunction and module in their study of ordered sets. Both notions (suitably adapted) are available in topology too, which permits us to investigate topological, metric and other kinds of spaces in a similar spirit. Therefore, relative to a choice $\Phi$ of modules, we consider spaces which admit all colimits with weight in $\Phi$, as well as (suitably defined) $\Phi$-distributive and $\Phi$-algebraic spaces. We show that the category of $\Phi$-distributive spaces and $\Phi$-colimit preserving maps is dually equivalent to the idempotent splitting completion of a category of spaces and convergence relations between them. We explain the connection of these results to the traditional duality of spaces with frames, and conclude further duality theorems. Finally, we study properties and structures of the resulting categories, in particular monoidal (closed) structures.

Our work is a fundamental study of the notion of approximation in V-categories and in (U,V)-categories, for a quantale V and the ultrafilter monad U. We introduce auxiliary, approximating and Scott-continuous distributors, the way-below distributor, and continuity of V- and (U,V)-categories. We fully characterize continuous V-categories (resp. (U,V)-categories) among all cocomplete V-categories (resp. (U,V)-categories) in the same ways as continuous domains are characterized among all dcpos. By varying the choice of the quantale V and the notion of ideals, and by further allowing the ultrafilter monad to act on the quantale, we obtain a flexible theory of continuity that applies to partial orders and to metric and topological spaces. We demonstrate on examples that our theory unifies some major approaches to quantitative domain theory., Comment: 17 pages

In the context of categorical topology, more precisely that of T-categories [Hofmann, 2007], we define the notion of T-colimit as a particular colimit in a V-category. A complete and cocomplete V-category in which limits distribute over T-colimits, is to be thought of as the generalisation of a (co-)frame to this categorical level. We explain some ideas on a T-categorical version of "Stone duality", and show that Cauchy completeness of a T-category is precisely its sobriety., Comment: 16 pages, modifications here and there, final version

Notions and techniques of enriched category theory can be used to study topological structures, like metric spaces, topological spaces and approach spaces, in the context of topological theories. Recently in [D. Hofmann, Injective spaces via adjunction, arXiv:0804.0326 [math.CV]] the construction of a Yoneda embedding allowed to identify injectivity of spaces as cocompleteness and to show monadicity of the category of injective spaces and left adjoints over $\mathsf{Set}$. In this paper we generalise these results, studying cocompleteness with respect to a given class of distributors. We show in particular that the description of several semantic domains presented in [M. Escard\'o and B. Flagg, Semantic domains, injective spaces and monads, Electronic Notes in Theoretical Computer Science 20 (1999)] can be translated into the $\mathsf{V}$-enriched setting.

Our work over the past years shows that not only the collection of (for instance) all topological spaces gives rise to a category, but also each topological space can be seen individually as a category by interpreting the convergence relation $\mathfrak{x}\to x$ between ultrafilters and points of a topological space $X$ as arrows in $X$. Naturally, this point of view opens the door to the use of concepts and ideas from (enriched) Category Theory for the investigation of (for instance) topological spaces. In this paper we study cocompleteness, adjoint functors and Kan extensions in the context of topological theories. We show that the cocomplete spaces are precisely the injective spaces, and they are algebras for a suitable monad on $\SET$. This way we obtain enriched versions of known results about injective topological spaces and continuous lattices., Comment: 24 pages

For a quantale $\V$, first a closure-theoretic approach to completeness and separation in $\V$-categories is presented. This approach is then generalized to $\Tth$-categories, where $\Tth$ is a topological theory that entails a set monad $\mT$ and a compatible $\mT$-algebra structure on $\V$.

It is known since 1973 that Lawvere's notion of (Cauchy-)complete enriched category is meaningful for metric spaces: it captures exactly Cauchy-complete metric spaces. In this paper we introduce the corresponding notion of Lawvere completeness for $(\mathbb{T},\mathsf{V})$-categories and show that it has an interesting meaning for topological spaces and quasi-uniform spaces: for the former ones means weak sobriety while for the latter means Cauchy completeness. Further, we show that $\mathsf{V}$ has a canonical $(\mathbb{T},\mathsf{V})$-category structure which plays a key role: it is Lawvere-complete under reasonable conditions on the setting; permits us to define a Yoneda embedding in the realm of $(\mathbb{T},\mathsf{V})$-categories.

Exponentiable functors between quantaloid-enriched categories are characterized in elementary terms. The proof goes as follows: the elementary conditions on a given functor translate into existence statements for certain adjoints that obey some lax commutativity; this, in turn, is precisely what is needed to prove the existence of partial products with that functor; so that the functor's exponentiability follows from the works of Niefield [1980] and Dyckhoff and Tholen [1987]., Comment: 10 pages; correction of flaw in proof