arXiv subjects

Walter Tholen

Publications and source records attributed to Walter Tholen.

At least 19 recordsLinked to original sources

arXiv2026

From Grothendieck cofibrations to factorization systems: a formal 2-monadic account

Grothendieck cofibrations describe transport in a category varying over a base, while factorization systems organize the arrows of a category into two complementary classes. We give a fully 2-categorical account of the passage from the former structure to the latter. The global comma 2-monad on the arrow 2-category encodes Grothendieck transport, whereas the squaring 2-monad encodes factorizations. We prove that split cofibrations are precisely the strict algebras for the comma 2-monad, including their 1-cells and 2-cells, and that normally cloven cofibrations are precisely its normal pseudoalgebras. A canonical colax morphism from the comma 2-monad to the squaring 2-monad then turns cocartesian transport into the cocartesian-vertical factorization of arrows in the total category. At the strict level, this yields the strict factorization system of designated cocartesian and vertical arrows; at the coherent level, it yields the orthogonal factorization system whose left class consists of all cocartesian arrows and whose right class consists of the arrows sent to isomorphisms in the base. We also separate unrestricted global pseudoalgebras, which retain a coherently trivial base action, from fixed-base pseudoalgebras, which correspond to arbitrary cleavages, and record the dual strict result for fibrations. This places the classical cofibration-factorization interaction, in all these variants, within a single change-of-2-monads construction and relates it directly to the existing fibrational and factorization literature.

math.CT↗

arXiv2026

Cauchy convergence in V-normed categories

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.

math.CT↗

arXiv2026

Revisiting Hugo Volger's paper Uber die Existenz der freien Algebren

We give a modern account of Hugo Volger's 1967 paper which, motivated by the construction of free algebras for a Lawvere-Linton theory, gives a very constructive proof that the left Kan extension of a product-preserving Set-valued functor is product-preserving. We also analyze how it anticipates, and in part even exceeds, subsequent work of the 1970s.

math.CT↗

arXiv2025

Categories of Games and their Fraïssé Theory

Relying on recent generalizations of the Fraïssé theory to a broader category-theoretic context, we study the class of abstract finite games played between two players and show the existence of an infinitetly countable game which is ultrahomogeneous and universal with respect to said class. Certain peculiarities of our game categories which clash with the usual framework found in the literature then lead us to formulate weaker category-theoretic properties which still yield a universal and ultrahomogeneous Fraïssé limit, thus further generalizing the categorical framework for a Fraïssé theory.

math.GM↗

arXiv2025

Infinitely ludic categories

Pursuing a new approach to the study of infinite games in combinatorics, we introduce the categories $\mathbf{Game}_{A}$ and $\mathbf{Game}_{B}$ and improve some classical results concerning topological games related to the duality between covering properties of $X$ and convergence properties of $\mathrm{C}_{\mathrm {p}}(X)$ by establishing the existence and key role of certain natural transformations. We then describe these ludic categories in various equivalent forms, viewing their objects as certain structured trees, presheaves, or metric spaces, and we thereby obtain their arboreal, functorial and metrical appearances. We use their metrical disguise to demonstrate a universality property of the Banach-Mazur game. The various equivalent descriptions come with underlying functors to more familiar categories which help establishing some important properties of the game categories: they are complete, cocomplete, extensive, cartesian closed, and coregular, but neither regular nor locally cartesian closed. We prove that their classes of strong epimorphisms, of regular epimorphisms, and of descent morphisms, are all distinct, and we show that these categories have weak classifiers for strong partial maps. Some of the categorical constructions have interesting game-theoretic interpretations.

math.GN↗

arXiv2025

Notions of Cauchy completeness for normed categories

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 idempotent-split property of an ordinary category when the governing symmetric monoidal-closed category is changed from the extended real half-line to the category of sets. In this paper, for any commutative quantale $\mathcal{V}$, we extend these two characterisations of Lawvere-style completeness to $\mathcal{V}$-normed categories, thus replacing $[0,\infty]$ and $\mathsf{Set}$ more generally by the category $\mathsf{Set}{/\!\!/}\mathcal{V}$ of $\mathcal{V}$-normed sets. We also establish improvements of recent results regarding the normed convergence of Cauchy sequences in two important $\mathcal{V}$-normed categories.

math.CT↗

arXiv2024

The normal decomposition of a morphism in categories without zeros

For a morphism f in a category C with sufficiently many finite limits and colimits, we discuss an elementary construction of a decomposition of f through objects P and N which, if C happens to have a zero object, amounts to the standard decomposition of f through P = Coker(ker f) and N = Ker(coker f). In this way we obtain natural notions of normal monomorphism and normal epimorphism also in non-pointed categories, as special types of regular mono- and epimorphisms. We examine the factorization behaviour of these classes of morphisms in general, compare the generalized normal decompositions with other types of threefold factorizations, and illustrate them in some every-day categories. The concrete construction of normal decompositions in the slices or coslices of these categories can be challenging. Amongst many others, in this regard, we consider particularly the categories of T1-spaces and of groups.

math.CT↗

arXiv2023

Groupoids and skeletal categories form a pretorsion theory in $\mathsf{Cat}$

We describe a pretorsion theory in the category $Cat$ of small categories: the torsion objects are the groupoids, while the torsion-free objects are the skeletal categories, i.e., those categories in which every isomorphism is an automorphism. We infer these results from two unexpected properties of coequalizers in $Cat$ that identify pairs of objects: they are faithful and reflect isomorphisms.

math.CT↗

arXiv2023

Smallness in Topology

Quillen's notion of small object and the Gabriel-Ulmer notion of finitely presentable or generated object are fundamental in homotopy theory and categorical algebra. Do these notions always lead to rather uninteresting classes of objects in categories of topological spaces, such as all finite discrete spaces, or just the empty space, as the examples and remarks in the existing literature may suggest? This article demonstrates that the establishment of full characterizations of these notions (and some natural variations thereof) in many familiar categories of spaces can be quite challenging and may lead to unexpected surprises. In fact, we show that there are significant differences in this regard even amongst the categories defined by the standard separation axioms, with the T1-separation condition standing out. The findings about these specific categories lead us to insights also when considering rather arbitrary full reflective subcategories of the category of all topological spaces.

math.GN↗

arXiv2022

A categorical review of complete regularity

We use the ultrafilter-convergence axiomatics for topological spaces to motivate in detail a gentle categorical introduction, first to Barr's Set-based relational T-algebras, and then to Burroni's T-preorders internal to a category C, here called T-spaces in C, for a monad T on C that substitutes the ultrafilter monad on Set. Within these settings one finds not only the notions of compactness and Hausdorff separation, originally due to Manes, but also that of complete regularity. Based on a somewhat hidden result by Burroni, the main theorem of this paper establishes an external fibrational characterization of the category of completely regular T-spaces with its reflexive subcategory of compact Hausdorff T-spaces, under modest assumptions on C and T.

math.GN↗

arXiv2021

Quotients of span categories that are allegories and the representation of regular categories

We consider the ordinary category Span(C) of (isomorphism classes of) spans of morphisms in a category C with finite limits as needed, composed horizontally via pullback, and give a general criterion for a quotient of Span(C) to be an allegory. In particular, when C carries a pullback-stable, but not necessarily proper, (E, M)-factorization system, we establish a quotient category Span_E(C) that is isomorphic to the category Rel_M(C) of M-relations in C, and show that it is a (unitary and tabular) allegory precisely when M is a class of monomorphisms in C. Without this restriction, one can still find a least pullback-stable and composition-closed class E. containing E such that Span_E.(C) is a unitary and tabular allegory. In this way one obtains a left adjoint to the 2-functor that assigns to every unitary and tabular allegory the regular category of its Lawverian maps. With the Freyd-Scedrov Representation Theorem for regular categories, we conclude that every finitely complete category with a stable factorization system has a reflection into the huge 2-category of all regular categories.

math.CT↗

arXiv2021

Diagrams, Fibrations, and the Decomposition of Colimits

The contributions of this paper are twofold. Within the framework of Grothendieck's fibrational category theory, we present a web of fundamental 2-adjunctions surrounding the formation of the category of all small diagrams in a given category and the formation of the Grothendieck category of a functor into the category of small categories. We demonstrate the utility of these adjunctions, in part by deriving three formulae for (co-)limits: a `twisted' generalization of the well-known Fubini formula, as first established by Chachólski and Scherer; a new `general colimit decomposition formula'; and a special case of the general formula, which actually initiated this work, and which was proved independently by Batanin and Berger. We give three proofs for this colimit decomposition formula, using methods that provide quite distinct insights. The `base' of our web of 2-adjunctions extends earlier work of the Ehresmann school and Guitart and promises to be of independent interest. It involves forming the diagram category of an arbitrary functor, seen as an object of the arrow category of the category of locally small categories, rather than that of a mere category. The left adjoint of the emerging generalized Guitart 2-adjunction factors through the 2-equivalence of split Grothendieck (co-)fibrations and strictly (co-)indexed categories, which we present here most generally by allowing 2-dimensional variation in the base categories.

math.CT↗

arXiv2021

Kan extensions are partial colimits

One way of interpreting a left Kan extension is as taking a kind of "partial colimit", whereby one replaces parts of a diagram by their colimits. We make this intuition precise by means of the "partial evaluations" sitting in the so-called bar construction of monads. The (pseudo)monads of interest for forming colimits are the monad of diagrams and the monad of small presheaves, both on the (huge) category CAT of locally small categories. Throughout, particular care is taken to handle size issues, which are notoriously delicate in the context of free cocompletion. We spell out, with all 2-dimensional details, the structure maps of these pseudomonads. Then, based on a detailed general proof of how the "restriction-of-scalars" construction of monads extends to the case of pseudoalgebras over pseudomonads, we define a morphism of monads between them, which we call "image". This morphism allows us in particular to generalize the idea of "confinal functors" i.e. of functors which leave colimits invariant in an absolute way. This generalization includes the concept of absolute colimit as a special case. The main result of this paper spells out how a pointwise left Kan extension of a diagram corresponds precisely to a partial evaluation of its colimit. This categorical result is analogous to what happens in the case of probability monads, where a conditional expectation of a random variable corresponds to a partial evaluation of its center of mass.

math.CT↗

arXiv2020

The comprehensive factorization of Burroni's T-functors

Expanding on the comprehensive factorization of functors internal to a category C, under fairly mild conditions on a monad T on C we establish that this orthogonal factorization system exists even in Burroni's category Cat(T) of (internal) T-categories and their functors. This context provides for some expected applications and some unexpected connections. For example, it lets us deduce that the comprehensive factorization is also available for functors of Lambek's multicategories. In topology, it leads to the insight that the role of discrete cofibrations is played by perfect maps, with the comprehensive factorization of a continuous map given by its fibrewise compactification.

math.CT↗

arXiv2019

Order-enriched solid functors

Order-enriched solid functors, as presented in this paper in two versions, enjoy many of the strong properties of their ordinary counterparts, including the transfer of the existence of weighted (co)limits from their codomains to their domains. The ordinary version of the notion first appeared in Trnková's work on automata theory of the 1970s and was subsequently studied by others under various names, before being put into a general enriched context by Anghel. Our focus in this paper is on differentiating the order-enriched notion from the ordinary one, mostly in terms of the functor's behaviour with respect to specific weighted (co)limits, and on the presentation of examples, which include functors of general varieties of ordered algebras and special ones, such as ordered vector spaces.

math.CT↗

arXiv2019

Metagories

Metric approximate categories, or metagories, for short, are metrically enriched graphs. Their structure assigns to every directed triangle in the graph a value which may be interpreted as the area of the triangle; alternatively, as the distance of a pair of consecutive arrows to any potential candidate for their composite. These values may live in an arbitrary commutative quantale. Generalizing and extending recent work by Aliouche and Simpson, we give a condition for the existence of an Yoneda-type embedding which, in particular, gives the isometric embeddability of a metagory into a metrically enriched category. The generality of the value quantale allows for applications beyond the classical metric context.

math.CT↗

arXiv2017

A Note on the Topologicity of Quantale-Valued Topological Spaces

For a quantale ${\sf{V}}$, the category $\sf V$-${\bf Top}$ of ${\sf{V}}$-valued topological spaces may be introduced as a full subcategory of those ${\sf{V}}$-valued closure spaces whose closure operation preserves finite joins. In generalization of Barr's characterization of topological spaces as the lax algebras of a lax extension of the ultrafilter monad from maps to relations of sets, for ${\sf{V}}$ completely distributive, ${\sf{V}}$-topological spaces have recently been shown to be characterizable by a lax extension of the ultrafilter monad to ${\sf{V}}$-valued relations. As a consequence, ${\sf{V}}$-$\bf Top$ is seen to be a topological category over $\bf Set$, provided that ${\sf{V}}$ is completely distributive. In this paper we give a choice-free proof that ${\sf{V}}$-$\bf Top$ is a topological category over $\bf Set$ under the considerably milder provision that ${\sf{V}}$ be a spatial coframe. When ${\sf{V}}$ is a continuous lattice, that provision yields complete distributivity of ${\sf{V}}$ in the constructive sense, hence also in the ordinary sense whenever the Axiom of Choice is granted.

cs.LO↗

arXiv2017

Lax distributive laws for topology, II

For a small quantaloid $\mathcal{Q}$ we consider four fundamental 2-monads $\mathbb{T}$ on $\mathcal{Q}\text{-}{\bf Cat}$, given by the presheaf 2-monad $\mathbb{P}$ and the copresheaf 2-monad $\mathbb{P}^{\dagger}$, as well as by their two composite 2-monads, and establish that they all laxly distribute over $\mathbb{P}$. These four 2-monads therefore admit lax extensions to the category $\mathcal{Q}\text{-}{\bf Dist}$ of $\mathcal{Q}$-categories and their distributors. We characterize the corresponding $(\mathbb{T},\mathcal{Q})$-categories in each of the four cases, leading us to both known and novel categorical structures.

math.CT↗