Related papers: On the categorical interpretation of ring cohomolo…
We study a categorical condition on relations, which is a categorical formulation of J\'onsson's characterisation of congruence distributive varieties. Categories satisfying these conditions need not be varieties; for instance, the dual of…
Starting from the observation that distinct notions of copying have arisen in different categorical fields (logic and computation, contrasted with quantum mechanics) this paper addresses the question of when, or whether, they may coincide.…
We show that the condition of being categorical in a tail of cardinals can be characterized algebraically for several classes of modules. $Theorem.$ Assume $R$ is an associative ring with unity. 1. The class of locally pure-injective…
To characterize categorical constraints - associativity, commutativity and monoidality - in the context of quasimonoidal categories, from a cohomological point of view, we define the notion of a parity (quasi)complex. Applied to groups…
We examine the consequences of having a total division operation $\frac{x}{y}$ on commutative rings. We consider two forms of binary division, one derived from a unary inverse, the other defined directly as a general operation; each are…
Every ring extension of $A$ by $R$ induces a pair of group homomorphisms $\mathcal{L}^{*}:R\to End_\Z(A)/L(A);\mathcal{R}^{*}:R\to End_\Z(A)/R(A),$ preserving multiplication, satisfying some certain conditions. A such 4-tuple…
Model theoretic internality provides conditions under which the group of automorphisms of a model over a reduct is itself a definable group. In this paper we formulate a categorical analogue of the condition of internality, and prove an…
Many of the properties of sectional category, topological complexity and homotopic distance are in fact derived from a small number of basic properties, which, once established, lead to all the others without further recourse to topology.…
The semantics of extensional type theory has an elegant categorical description: models of extensional =-types, 1-types, and Sigma-types are biequivalent to finitely complete categories, while adding Pi-types yields locally Cartesian closed…
Let $A$ be an algebra over a commutative ring $k$. We prove that braidings on the category of $A$-bimodules are in bijective correspondence to canonical R-matrices, these are elements in $A\ot A\ot A$ satisfying certain axioms. We show that…
In [FGRS1,FGRS2] the relationship between the universal and elementary theory of a group ring $R[G]$ and the corresponding universal and elementary theory of the associated group $G$ and ring $R$ was examined. Here we assume that $R$ is a…
Adhesive categories provide an abstract framework for the algebraic approach to rewriting theory, where many general results can be recast and uniformly proved. However, checking that a model satisfies the adhesivity properties is sometimes…
A commutative ring is reduced when it can be embedded into a direct product of fields. While the category of reduced commutative rings plays a fundamental role in affine geometry, it exhibits several structural deficiencies: it admits…
Starting from a generalization of the standard axioms for a monoid we present a stepwise development of various, mutually equivalent foundational axiom systems for category theory. Our axiom sets have been formalized in the Isabelle/HOL…
In arXiv:2511.04191 we constructed schemes of objects in small categories which contained a set of basepoints with local representing (localizing) objects. Here we prove that the category $\cat{Rings}$ of associative rings with unit has a…
We give a necessary and sufficient condition in terms of group cohomology for two indecomposable module categories over a group-theoretical fusion category ${\mathcal C}$ to be equivalent. This concludes the classification of such module…
Category theory unifies mathematical concepts, aiding comparisons across structures by incorporating objects and morphisms, which capture their interactions. It has influenced areas of computer science such as automata theory, functional…
We introduce an approach to the categorification of rings, via the notion of distributive categories with negative objects, and use it to lay down categorical foundations for the study of super, quantum and non-commutative combinatorics.…
Under mild assumptions, we construct the two Matlis additive category equivalences for an associative ring epimorphism $u\colon R\to U$. Assuming that the ring epimorphism is homological of flat/projective dimension $1$, we discuss the…
For any commutative ring $R$, we show that the categories of $R$-coalgebras and cocommutative $R$-coalgebras are locally $\aleph_1$-presentable, while the categories of $R$-flat $R$-coalgebras are $\aleph_1$-accessible. Similarly, for any…