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.