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Dirk Hofmann

Publications and source records attributed to Dirk Hofmann.

At least 37 records · Page 2Linked to original sources
arXiv2017

Limits in Categories of Vietoris Coalgebras

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.

cs.LO↗
arXiv2016

Continuity as a computational effect

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.

cs.LO↗
arXiv2015

A cottage industry of lax extensions

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.

math.CT↗
arXiv2014

Representable (T, V)-categories

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.

math.CT↗
arXiv2014

Some notes on Esakia spaces

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.

math.CT↗
arXiv2013

Measuring excitation-energy transfer with a real-time time-dependent density functional theory approach

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 Schrödinger 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.

cond-mat.mtrl-sci↗
arXiv2013

Exponential Kleisli monoids as Eilenberg-Moore algebras

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.

math.CT↗
arXiv2013

The enriched Vietoris monad on representable spaces

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 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.

math.CT↗
arXiv2013

Exponentiable approach spaces

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

math.GN↗
arXiv2013

Dualities in modal logic from the point of view of triples

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.

math.LO↗
arXiv2012

On a coalgebraic view on Logic

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.

math.LO↗
arXiv2011

Approaching metric domains

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.

math.GN↗
arXiv2011

A four for the price of one duality principle for distributive 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.

math.GN↗
arXiv2010

A Duality of Quantale-Enriched Categories

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.

math.CT↗
arXiv2010

Stone duality for topological theories

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.

math.CT↗
arXiv2010

Duality for distributive space

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 $Φ$ of modules, we consider spaces which admit all colimits with weight in $Φ$, as well as (suitably defined) $Φ$-distributive and $Φ$-algebraic spaces. We show that the category of $Φ$-distributive spaces and $Φ$-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.

math.CT↗
arXiv2010

Approximation in quantale-enriched categories

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.

math.CT↗