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Walter Tholen

Publications and source records attributed to Walter Tholen.

29 records · Page 2Linked to original sources
arXiv2016

Approximate injectivity

In a locally $λ$-presentable category, with $λ$ a regular cardinal, classes of objects that are injective with respect to a family of morphisms whose domains and codomains are $λ$-presentable, are known to be characterized by their closure under products, $λ$-directed colimits and $λ$-pure subobjects. Replacing the strict commutativity of diagrams by "commutativity up to $\varepsilon$", this paper provides an "approximate version" of this characterization for categories enriched over metric spaces. It entails a detailed discussion of the needed $\varepsilon$-generalizations of the notion of $λ$-purity. The categorical theory is being applied to the locally $\aleph_1$-presentable category of Banach spaces and their linear operators of norm at most 1, culminating in a largely categorical proof for the existence of the so-called Gurarii Banach space.

math.CT↗
arXiv2016

Monads on Q-Cat and their lax extensions to Q-Dist

For a small quantaloid $\mathcal{Q}$, we consider 2-monads on the 2-category $\mathcal{Q}$-$\bf{Cat}$ and their lax extensions to the 2-category $\mathcal{Q}$-$\bf{Dist}$ of small $\mathcal{Q}$-categories and their distributors, in particular those lax extensions that are flat, in the sense that they map identity distributors to identity distributors. In fact, unlike in the discrete case, a 2-monad on $\mathcal{Q}$-$\bf{Cat}$ may admit only one flat lax extension. Every ordinary monad on the comma category $\bf{Set}/{\rm ob}\mathcal{Q}$ with a lax extension to $\mathcal{Q}$-$\bf{Rel}$ gives rise to such a 2-monad on $\mathcal{Q}$-$\bf{Cat}$, and we describe this process globally as a coreflective embedding. The $\mathcal{Q}$-presheaf and the double $\mathcal{Q}$-presheaf monads are important examples of 2-monads on $\mathcal{Q}$-$\bf{Cat}$ allowing flat lax extensions to $\mathcal{Q}$-$\bf{Dist}$, and so are their submonads, obtained by the restriction to conical (co)presheaves and known as the $\mathcal{Q}$-Hausdorff and double $\mathcal{Q}$-Hausdorff monads, which we define here in full generality, thus generalizing some previous work in the case when $\mathcal{Q}$ is a quantale, or just the "metric" quantale $[0,\infty]$. Their discretization leads naturally to various lax extensions of the relevant $\bf{Set}$-monads used in monoidal topology.

math.CT↗
arXiv2016

Lax Distributive Laws for Topology, I

For a quantaloid $\mathcal{Q}$, considered as a bicategory, Walters introduced categories enriched in $\mathcal{Q}$. Here we extend the study of monad-quantale-enriched categories of the past fifteen years by introducing monad-quantaloid-enriched categories. We do so by making lax distributive laws of a monad $\mathbb{T}$ over the discrete presheaf monad of the small quantaloid $\mathcal{Q}$ the primary data of the theory, rather than the lax monad extensions of $\mathbb{T}$ to the category of $\mathcal{Q}$-relations that they equivalently describe. The central piece of the paper establishes a Galois correspondence between such lax distributive laws and lax Eilenberg-Moore $\mathbb{T}$-algebra structures on the set of discrete presheaves over the object set of $\mathcal{Q}$. We give a precise comparison of these structures with the more restrictive notion introduced by Hofmann in the case of a commutative quantale, called natural topological theories here, and describe the lax monad extensions introduced by him as minimal. Throughout the paper, a variety of old and new examples of ordered, metric and topological structures illustrate the theory developed, which includes the consideration of algebraic functors and change-of-base functors in full generality.

math.CT↗
arXiv2016

Quantale-valued Approach Spaces via Closure and Convergence

For a quantale $\mathsf{V}$ we introduce $\mathsf{V}$-approach spaces via $\mathsf{V}$-valued point-set-distance functions and, when $\mathsf{V}$ is completely distributive, characterize them in terms of both, so-called closure towers and ultrafilter convergence relations. When $\mathsf{V}$ is the two-element chain $\mathsf{2}$, the extended real half-line $[0,\infty]$, or the quantale ${\bfΔ}$ of distance distribution functions, the general setting produces known and new results on topological spaces, approach spaces, and the only recently considered probabilistic approach spaces, as well as on their functorial interactions with each other.

math.GN↗
arXiv2016

Topological categories, quantaloids and Isbell adjunctions

In fairly elementary terms this paper presents, and expands upon, a recent result by Garner by which the notion of topologicity of a concrete functor is subsumed under the concept of total cocompleteness of enriched category theory. Motivated by some key results of the 1970s, the paper develops all needed ingredients from the theory of quantaloids in order to place essential results of categorical topology into the context of quantaloid-enriched category theory, a field that previously drew its motivation and applications from other domains, such as quantum logic and sheaf theory.

math.CT↗
arXiv2015

Categorically proper homomorphisms of topological groups

We extend the Dikranjan-Uspenskij notions of c-compact and h-complete topological group to the morphism level, study the stability properties of the newly defined types of maps, such as closure under direct products, and compare them with their counterparts in topology. We assume Hausdorffness only when our proofs require us to do so, which leads to new results and the affirmation of some facts that were known in a Hausdorff context.

math.GN↗
arXiv2015

The fundamental group as the structure of a dually affine space

This paper dualizes the setting of affine spaces as originally introduced by Diers for application to algebraic geometry and expanded upon by various authors, to show that the fundamental groups of pointed topological spaces appear as the structures of dually affine spaces. The dual of the Zariski closure operator is introduced, and the 1-sphere and its copowers together with their fundamental groups are shown to be examples of complete objects with respect to the Zariski dual closure operator.

math.CT↗
arXiv2009

On the categorical meaning of Hausdorff and Gromov distances, I

Hausdorff and Gromov distances are introduced and treated in the context of categories enriched over a commutative unital quantale V. The Hausdorff functor which, for every V-category X, provides the powerset of X with a suitable V-category structure, is part of a monad on V-Cat whose Eilenberg-Moore algebras are order-complete. The Gromov construction may be pursued for any endofunctor K of V-Cat. In order to define the Gromov "distance" between V-categories X and Y we use V-modules between X and Y, rather than V-category structures on the disjoint union of X and Y. Hence, we first provide a general extension theorem which, for any K, yields a lax extension K to the category V-Mod of V-categories, with V-modules as morphisms.

math.CT↗
arXiv2007

Lawvere completion and separation via closure

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 $\V$.

math.CT↗
arXiv2007

Factorization, Fibration and Torsion

A simple definition of torsion theory is presented, as a factorization system with both classes satisfying the 3--for--2 property. Comparisons with the traditional notion are given, as well as connections with the notions of fibration and of weak factorization system, as used in abstract homotopy theory.

math.AT↗