Exact sequence of monoids

Hi John. I'd say there is no generalization of short exact sequence to the category of monoids, although I suppose it really depends on what you want to do with it. What you probably want is an internal equivalence relation. So you could say a diagram $A\rightrightarrows B \to C$ of monoid maps (where the two compositions $A\rightrightarrows C$ agree) is short exact if the map $B\to C$ is surjective and the induced map $A\to B\times_C B$ is an isomorphism. This is equivalent to requiring the induced map $A\to B\times B$ to be injective, its image to be an equivalence relation, and the induced map $B/A\to C$ is to be an isomorphism.

My general feeling is that this is the right concept in most categories of sets with algebraic structure (e.g. the category of sets itself, semi-rings). It's only in categories where the objects have some group structure that you can re-express it using kernels.

For the special case of commutative monoids, or more generally semimodules over a semiring, my related preprints on arXiv might be helpful (see below). For arbitrary monoids, the general definition is similar, however it is difficult to apply since the notion of the cokernel of a morphism of monoids is really "slippery":

http://arxiv.org/abs/1111.0330

http://arxiv.org/abs/1210.4566