Re: Mike Barr's question concerning two equivalent definitions of a class of categories Since the term 'additive' had already been established to refer to the special case where the homs are abelian groups, I called these 'linear categories' in my paper Categories of Space and of Quantity in The Space of Mathematics, Philosophical, Epistemological and Historical Explorations, de Gruyter, Berlin (1992) pp 14-30 because 'linear' is a term well known to engineers, statisticians and others, and because these categories form the natural environment for applications of Linear Algebra. Of course, the entries in the matrices are in general maps, not necessarily scalars, although scalars for which the addition is idempotent are an important special case. (Here by the scalars of such a category I mean the elements of the rig which is its center.) In that paper I referred to what I believe is the first reference to this theory, namely Saunders Mac Lane's 1950 paper Duality for Groups, Bull AMS vol 56, pp 485-516, (1950) expounding work he did in the late 40's. Bill Lawvere ****************************************************************** F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA ****************************************************************** On Thu, 29 Oct 1998, Michael Barr wrote:

Can someone give me a reference for the fact that if the hom functor on a category factors through commutative monoids then finite products are sums and vice versa. Also conversely.

Michael