44 citations · 366 across the 37 of their papers we have counts for
28 papers · 1 filter
Observations on the variety of equationally linear Heyting semilattices
Xabier García-Martínez, James R. A. Gray, Michael A. Hoefnagel +2
In previous work, we analysed a number of categorical properties, of interest in the context of Janelidze-Márki-Tholen semi-abelian categories, for the variety of…
A two-categorical Snake Lemma
Elena Caviglia, Luca Mesiti, Tim Van der Linden
We prove a Snake Lemma for 2-categories. Working in a 2-category with a strong bizero object, we develop 2-kernels and 2-cokernels, 2-monomorphisms and 2-epimorphisms as fully fait…
Categorical-algebraic aspects of Heyting semilattices
Xabier García-Martínez, James R. A. Gray, Michael A. Hoefnagel +2
This article gives an overview of some key categorical-algebraic properties of the variety of Heyting semilattices, with the aim of correcting a misconception in the literature. We…
Non-additive derived functors via chain resolutions
Maxime Culot, Fara Renaud, Tim Van der Linden
Let be a functor from a category to a homological (Borceux-Bourn) or semi-abelian (Janelidze-Márki-Tholen) category $\mathcal{E}…
A Homological View of Categorical Algebra
George Peschke, Tim Van der Linden
We provide a foundation for working with homological and homotopical methods in categorical algebra. This involves two mutually complementary components, namely (a) the strategic s…
Twisted Commutators and Internal Crossed Modules
Bo Shan Deval, Tim Van der Linden
We introduce a notion of relative commutator -- an important special case being commutators twisted by an action -- as a straightforward modification of the definition of the Higgi…
A universal Kaluzhnin--Krasner embedding theorem
Bo Shan Deval, Xabier García-Martínez, Tim Van der Linden
Given two groups and , the Kaluzhnin--Krasner universal embedding theorem states that the wreath product acts as a universal receptacle for extensions from to $…
Weak representability of actions of non-associative algebras
Jose Brox, Xabier García-Martínez, Manuel Mancini +2
We study the categorical-algebraic condition that internal actions are weakly representable (WRA) in the context of varieties of (non-associative) algebras over a field. Our first…
Homological lemmas for (non-abelian) group-like structures by diagram chasing in a self-dual context
Kishan Kumar Dayaram, Amartya Goswami, Zurab Janelidze +2
Through abelian categories, homological lemmas for modules admit a self-dual treatment, where half of the proof of a lemma is sufficient to prove the full lemma. In this paper, we…
A symmetric approach to higher coverings in categorical Galois theory
Fara Renaud, Tim Van der Linden
In the context of a tower of (strongly Birkhoff) Galois structures in the sense of categorical Galois theory, we show that the concept of a higher covering admits a characterisatio…
Algebraic logoi
D. Bourn, A. S. Cigoli, J. R. A. Gray +1
We introduce normal cores, as well as the more general action cores, in the context of a semi-abelian category, and further generalise those to split extension cores in the context…
A characterisation of Lie algebras via algebraic exponentiation
Xabier García-Martínez, Tim Van der Linden
In this article we describe varieties of Lie algebras via algebraic exponentiation, a concept introduced by Gray in his Ph.D. thesis. For an infinite field of characte…
Two characterisations of groups amongst monoids
Andrea Montoli, Diana Rodelo, Tim Van der Linden
The aim of this paper is to solve a problem proposed by Dominique Bourn: to provide a categorical-algebraic characterisation of groups amongst monoids and of rings amongst semiring…
What is an ideal a zero-class of?
N. Martins-Ferreira, A. Montoli, A. Ursini +1
We characterise, in pointed regular categories, the ideals as the zero-classes of surjective relations. Moreover, we study a variation of the "Smith is Huq" condition: two surjecti…
A criterion for reflectiveness of normal extensions
Andrea Montoli, Diana Rodelo, Tim Van der Linden
We give a new sufficient condition for the normal extensions in an admissible Galois structure to be reflective. We then show that this condition is indeed fulfilled when X is the…
Higher central extensions and cohomology
Diana Rodelo, Tim Van der Linden
We establish a Galois-theoretic interpretation of cohomology in semi-abelian categories: cohomology with trivial coefficients classifies central extensions, also in arbitrarily hig…
Algebraically coherent categories
Alan S. Cigoli, James R. A. Gray, Tim Van der Linden
We call a finitely complete category algebraically coherent when the change-of-base functors of its fibration of points are coherent, which means that they preserve finite limits a…
Peri-abelian categories and the universal central extension condition
James R. A. Gray, Tim Van der Linden
We study the relation between Bourn's notion of peri-abelian category and conditions involving the coincidence of the Smith, Huq and Higgins commutators. In particular we show that…
On the normality of Higgins commutators
Alan S. Cigoli, James R. A. Gray, Tim Van der Linden
In a semi-abelian context, we study the condition (NH) asking that Higgins commutators of normal subobjects are normal subobjects. We provide examples of categories that do or do n…
The fundamental group functor as a Kan extension
Tomas Everaert, Julia Goedecke, Tim Van der Linden
We prove that the fundamental group functor from categorical Galois theory may be computed as a Kan extension.
An observation on n-permutability
Nelson Martins-Ferreira, Diana Rodelo, Tim Van der Linden
We prove that in a regular category all reflexive and transitive relations are symmetric if and only if every internal category is an internal groupoid. In particular, these condit…
A decomposition formula for the weighted commutator
Nelson Martins-Ferreira, Tim Van der Linden
We decompose the weighted subobject commutator of M. Gran, G. Janelidze and A. Ursini as a join of a binary and a ternary commutator.
Universal central extensions in semi-abelian categories
Jose Manuel Casas, Tim Van der Linden
Basing ourselves on Janelidze and Kelly's general notion of central extension, we study universal central extensions in the context of semi-abelian categories. We consider a new fu…
Resolutions, higher extensions and the relative Mal'tsev axiom
Tomas Everaert, Julia Goedecke, Tim Van der Linden
We study how the concept of higher-dimensional extension which comes from categorical Galois theory relates to simplicial resolutions. For instance, an augmented simplicial object…