A Note on the Abelianization Functor

It is well known that the abelianization of a group G can be computed as the cokernel of the diagonal morphism (1_G, 1_G): G → G × G in the category of groups. We generalize this to arbitrary regular subtractive categories, among which are the category of groups, the category of topological groups, and the categories of other group-like structures. We also establish that an abelian category is the same as a regular subtractive category in which every monomorphism is a kernel of some morphism.