Reference?

Reference?

[Michael Barr]

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

Re: Reference?

[boerger]

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 

Re: Reference?

[F W Lawvere]

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
> 
> 
> 

Re: Reference?

[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