Two characterisations of groups amongst monoids

Abstract

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 semirings. In the case of monoids, our solution is given by the following equivalent conditions: (i) $G$ is a group; (ii) $G$ is a Mal'tsev object, i.e., the category of points over $G$ in the category of monoids is unital; (iii) $G$ is a protomodular object, i.e., all points over $G$ are stably strong. We similarly characterise rings in the category of semirings. On the way we develop a local or object-wise approach to certain important conditions occurring in categorical algebra. This leads to a basic theory involving what we call unital and strongly unital objects, subtractive objects, Mal'tsev objects and protomodular objects. We explore some of the connections between these new notions and give examples and counterexamples.