Related papers: Restriction categories as enriched categories
We combine the theory of inductive data types with the theory of universal measurings. By doing so, we find that many categories of algebras of endofunctors are actually enriched in the corresponding category of coalgebras of the same…
We extend the `join-premorphisms' part of the Ehresmann-Schein-Nambooripad Theorem to the case of two-sided restriction semigroups and inductive categories, following on from a result of Lawson (1991) for the `morphisms' part. However, it…
This is a companion overview paper to arXiv:2409.17489: we give all the main definitions, constructions and statements, but no proofs.
Motivated by the definition of Freiman homomorphism, we explore the possibilities of formulating some basic notions and techniques of additive combinatorics in a categorical language. We show that additive sets and Freiman homomorphisms…
Category theory is the language of homological algebra, allowing us to state broadly applicable theorems and results without needing to specify the details for every instance of analogous objects. However, authors often stray from the realm…
We study collections of additive categories $\mathcal{M}(G)$, indexed by finite groups $G$ and related by induction and restriction in a way that categorifies usual Mackey functors. We call them `Mackey 2-functors'. We provide a large…
A finite EI category is a small category with finitely many morphisms such that every endomorphism is an isomorphism. They include finite groups, finite posets and free categories of finite quivers as special cases. In this paper we…
The purpose of the paper is to rectify a series of errors occurred in [2], [17], [20] for a particular situation. To get a fruitful solution and to overcome the issue, we introduce a new form of set sharing namely restricted set sharing,…
We extend the framework of combinatorial model categories, so that the category of small presheaves over large indexing categories and ind-categories would be embraced by the new machinery called class-combinatorial model categories. The…
This is a write-up of a talk given at the CATMI meeting in Bergen in July 2023, and is an introduction to a category-theoretic perspective on metric spaces. A metric space is a set of points such that between each pair of points there is a…
We show how an effect algebra $\mathcal{X}$ can be regarded as a category, where the morphisms $x \rightarrow y$ are the elements $f$ such that $x \leq f \leq y$. This gives an embedding $\mathbf{EA} \rightarrow \mathbf{Cat}$. The interval…
We generalise to a group homomorphism $\tau$ the $\chi$-graded categories of S\"{o}zer and Virelizier. These are categories in which both morphisms and objects have compatible degrees. We give a 'half-enriched' Yoneda lemma, a structure…
We consider three (2-)categories and their (anti-)equivalence. They are the category of small abelian categories and exact functors, the category of definable additive categories and interpretation functors, the category of locally coherent…
Expansions of abelian categories are introduced. These are certain functors between abelian categories and provide a tool for induction/reduction arguments. Expansions arise naturally in the study of coherent sheaves on weighted projective…
We introduce the category of finite strings and study its basic properties. The category is closely related to the augmented simplex category, and it models categories of linear representations. Each lattice of non-crossing partitions…
Additive categories play a fundamental role in mathematics and related disciplines. Given an additive category equipped with a biadditive functor, one can construct its category of extensions, which encodes important structural information.…
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…
We construct a model structure on the category of small categories enriched over a combinatorial closed symmetric monoidal model category satisfying the monoid axiom. Weak equivalences are Dwyer-Kan equivalences, i.e. enriched functors…
The feasibility of a classification-by-rank program for modular categories follows from the Rank-Finiteness Theorem. We develop arithmetic, representation theoretic and algebraic methods for classifying modular categories by rank. As an…
We develop the theory of approximate Fra\"{i}ss\'{e} limits in the context of categories enriched over metric spaces. Among applications, we construct a generic projection on the Gurarii space and we present a simpler proof of a…