Abstract 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 $$\mathcal {V}$$ V taking the role of $$[0,\infty ]$$ [ 0 , ∞ ] , we extend these two characterisations of Lawvere-style completeness from ordinary to $$\mathcal {V}$$ V -normed categories, that is, to categories enriched in the category of $$\mathcal {V}$$ V -normed sets, i.e., of sets equipped with a $$\mathcal {V}$$ V -valued function. We also establish improvements of recent results regarding the normed convergence of Cauchy sequences in two important $$\mathcal {V}$$ V -normed categories.

Abstract A $$\mathcal {C}-\text {set}$$ C - set is a functor from a category $$\mathcal {C}$$ C to the category of finite sets, and $$\mathcal {C}-\text {set}$$ C - set denotes the category of such functors with natural transformations as morphisms. In this work, we prove that $$\mathcal {C}$$ C is a groupoid if and only if $$\mathcal {C}-\text {set}$$ C - set has finitely many indecomposable objects, which is the categorical analog of transitive G -sets in classical group actions.

Abstract What is a time-varying graph, a time-varying topological space, or, more generally, a mathematical structure that evolves over time? In this work, we lay the foundations for a general theory of temporal data by introducing categories of narratives . These are sheaves on posets of time intervals that encode snapshots of a temporal object along with the relationships between them. This theory satisfies five desiderata distilled from the burgeoning field of time-varying graphs: (D1) it defines both time-varying objects and their morphisms; (D2) it distinguishes between cumulative and persistent interpretations and provides principled methods for transitioning between them; (D3) it systematically lifts static notions to their temporal analogues; (D4) it is object agnostic; (D5) it integrates with theories of dynamical systems. To achieve this, we build upon existing categorical and sheaf-theoretic approaches to temporal graph theory, generalizing them to any category with limits and colimits. We also formalize tacit intuitions that, while present, often remain implicit in temporal graph theory. Beyond synthesizing and reformulating existing ideas in categorical language, we introduce sheaf-theoretic constructions and prove results that, to our knowledge, have not appeared in the temporal data literature—such as the adjunction between persistent and cumulative narratives. More importantly, we integrate these existing and novel elements into a consistent and coherent framework, setting the stage for a unified theory of time-varying data.

Abstract We use the theory of approximable triangulated categories to give a condition for a proper DG-category to be reflexive in the sense of Kuznetsov and Shinder. To do this we provide another description of the completion of an approximable triangulated category under a properness assumption. We apply our results to proper schemes, proper connective DG-algebras and Azumaya algebras over proper schemes. We include an appendix by Raedschelders and Stevenson showing that proper connective DG-algebras admit finite dimensional models over any field.

Abstract 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 form a category and we study several limit and colimit constructions in this category and in one of its interesting subcategories. Moreover, we study the additive structure of these (co)limit objects using the additive doubling constant. We relate this category to that of finite sets and mappings, and to that of abelian groups and group homomorphisms. We show that the Konyagin and Lev result on the existence of universal ambient groups is an instance of an adjunction.