Homology, Homotopy and Applications

Volume 18 (2016)

Number 1

Monoids and pointed $S$-protomodular categories

Pages: 151 – 172

DOI: https://dx.doi.org/10.4310/HHA.2016.v18.n1.a9

Authors

Dominique Bourn (Laboratoire de Mathématiques pures et appliquées, Université du Littoral Côte d’Opale, Calais, France)

Nelson Martins-Ferreira (ESTG, CDRSP, Instituto Politécnico de Leiria, Portugal)

Andrea Montoli (CMUC, University of Coimbra, Portugal)

Manuela Sobral (CMUC and Department of Mathematics, University of Coimbra, Portugal)

Abstract

We investigate the notion of pointed $S$-protomodular category, with respect to a suitable class $S$ of points, and we prove that these categories satisfy, relatively to the class $S$, many partial aspects of the properties of Mal’tsev and protomodular categories, like the split short five lemma for $S$-split exact sequences, or the fact that a reflexive $S$-relation is transitive. The main examples of $S$-protomodular categories are the category of monoids and, more generally, any category of monoids with operations, where the class $S$ is the class of Schreier points.

Keywords

fibration of points, Mal’tsev and protomodular categories, monoid with operations, Schreier split epimorphism, pointed $S$-protomodular category

2010 Mathematics Subject Classification

03C05, 08C05, 18D35, 18G50

Published 31 May 2016