Date: Thu, 29 Oct 1998 14:00:03 -0500 (EST) From: Michael Barr Subject: categories: Reference? 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 Subject: Re: categories: Reference? Date: Fri, 30 Oct 1998 09:26:24 +0000 (GMT) From: "Dr. P.T. Johnstone" > > 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 Mac Lane (Categories for the Working Mathematician) does one direction in Theorem 2 on page 190. (He assumes enrichment over abelian groups rather than commutative monoids, but a glance at the proof shows that the additive inverses are not used.) The converse is stated as Exercise 4 on page 194. Peter Johnstone Date: Fri, 30 Oct 1998 11:02:31 -0500 (EST) From: F W Lawvere Subject: categories: Re: Reference? 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 > > > Date: Fri, 30 Oct 1998 15:50:46 -0500 (EST) From: Michael Barr Subject: categories: Linear categories I have checked Mac Lane's 1950 paper and I cannot find any such result. The converse, that a category in which finite sums are equivalent to products then the category takes homs in commutative monoids is sort of there, but the one I asked is not, or at least I didn't find it. However, CWM is a likely source and I will check that out. I just need some reference in any case. Michael Date: Sat, 31 Oct 1998 14:19:05 -0500 (EST) From: F W Lawvere Subject: categories: Re: Linear categories Dear Mike and everbody The converse result , as stated by Mac Lane, was what needed to be said in 1950, especially since it began to bring out that category theory has content. What it's converse to, namely that when maps can be added then the cartesian product has the mapping property now called coproduct, had already been folklore for years... or if it hadn't been, how else to explain the widespread terminological ambiguity concerning "direct sums"? Perhaps a much older reference needs to be adduced. Thanks for bringing this up. Best regards Bill ******************************************************************************* F. William Lawvere Mathematics Dept. SUNY wlawvere@acsu.buffalo.edu 106 Diefendorf Hall 716-829-2144 ext. 117 Buffalo, N.Y. 14214, USA ******************************************************************************* On Fri, 30 Oct 1998, Michael Barr wrote: > I have checked Mac Lane's 1950 paper and I cannot find any such result. > The converse, that a category in which finite sums are equivalent to > products then the category takes homs in commutative monoids is sort of > there, but the one I asked is not, or at least I didn't find it. However, > CWM is a likely source and I will check that out. I just need some > reference in any case. > > Michael > > > From: boerger Date: Tue, 3 Nov 1998 12:19:11 +0100 Subject: Re: categories: Reference? The result that finite products coincide with finite coproducts in categories enriched commutative monoids can be found in Herrlich`s and Strecker`s book under 40.8 (p.308 in the 2nd edition). The converse is given there under 4.12 (p.310). They use the term "semi-additive category", which I am also used to. Though I agree with Bill Lawvere that prefixes like "semi" should be omitted if possible, I am not convinced by his suggestion "linear categeory" because for me subtraction seems essential fo linear algebra. Maybe somebody invents a better term. Greetings Reinhard