12 citations · 60 across the 29 of their papers we have counts for
28 papers · 1 filter
Notions of Cauchy completeness for normed categories
Dirk Hofmann, Walter Tholen
As already mentioned by Lawvere in his 1973 paper, the characterisation of Cauchy completeness of metric spaces in terms of representability of adjoint distributors amounts to the…
Topological lax comma categories
Maria Manuel Clementino, Dirk Hofmann, Rui Prezado
This paper investigates the interplay between properties of a topological space , in particular of its natural order, and properties of the lax comma category $\mathsf{Top} \Dow…
Barr-coexactness for metric compact Hausdorff spaces
Marco Abbadini, Dirk Hofmann
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…
A variety of co-quasivarieties
Maria Manuel Clementino, Carlos Fitas, Dirk Hofmann
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 sp…
Cauchy convergence in V-normed categories
Maria Manuel Clementino, Dirk Hofmann, Walter Tholen
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…
Kantorovich Functors and Characteristic Logics for Behavioural Distances
Sergey Goncharov, Dirk Hofmann, Pedro Nora +2
Behavioural distances measure the deviation between states in quantitative systems, such as probabilistic or weighted systems. There is growing interest in generic approaches to be…
A Point-free Perspective on Lax extensions and Predicate liftings
Sergey Goncharov, Dirk Hofmann, Pedro Nora +2
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 mod…
Duality theory for enriched Priestley spaces
Dirk Hofmann, Pedro Nora
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 struct…
Hausdorff coalgebras
Dirk Hofmann, Pedro Nora
As composites of constant, (co)product, identity, and powerset functors, Kripke polynomial functors form a relevant class of -functors in the theory of coalgebras. Th…
Cartesian closed exact completions in topology
Maria Manuel Clementino, Dirk Hofmann, Willian Ribeiro
Using generalized enriched categories, in this paper we show that Rosický's proof of cartesian closedness of the exact completion of the category of topological spaces can be exten…
Convergence and quantale-enriched categories
Dirk Hofmann, Carla Reis
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 th…
Aspects of algebraic Algebras
Dirk Hofmann, Lurdes Sousa
In this paper we investigate important categories lying strictly between the Kleisli category and the Eilenberg-Moore category, for a Kock-Zöberlein monad on an order-enriched cate…
Topology from enrichment: the curious case of partial metrics
Dirk Hofmann, Isar Stubbe
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 inter…
Enriched Stone-type dualities
Dirk Hofmann, Pedro Nora
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 o…
A cottage industry of lax extensions
Dirk Hofmann, Gavin J. Seal
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…
Representable (T, V)-categories
Dimitri Chikhladze, Maria Manuel Clementino, Dirk Hofmann
Working in the framework of -categories, for a symmetric monoidal closed category and a (not necessarily cartesian) monad , we present a common account to the study…
Some notes on Esakia spaces
Dirk Hofmann, Pedro Nora
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…
The enriched Vietoris monad on representable spaces
Dirk Hofmann
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…
Exponential Kleisli monoids as Eilenberg-Moore algebras
Dirk Hofmann, Frédéric Mynard, Gavin J. Seal
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 identifie…
A Duality of Quantale-Enriched Categories
Dirk Hofmann, Pawel Waszkiewicz
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…
Duality for distributive space
Dirk Hofmann
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…
Approximation in quantale-enriched categories
Dirk Hofmann, Pawel Waszkiewicz
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, approx…
Stone duality for topological theories
Dirk Hofmann, Isar Stubbe
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 a…
Relative injectivity as cocompleteness for a class of distributors
Maria Manuel Clementino, Dirk Hofmann
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 topolo…