Subject: categories: tensor product of categories
From: john baez <[email-masked]>
Date: Tue, 24 Mar 1998 12:25:34 -0800 (PST)

Where can I read about the "tensor product" of cocomplete categories?
(Hopefully this is a sensible and self-explanatory concept.)
Or variations on this theme involving categories with coproducts,
or finite colimits, or finite coproducts?  

Date: Thu, 26 Mar 1998 15:27:09 +1100 (EST)
From: [email-masked] (Max Kelly)
Subject: categories: Re: tensor product of categories

This is in response to John Baez' query:

Where can I read about the "tensor product" of cocomplete categories?
(Hopefully this is a sensible and self-explanatory concept.)
Or variations on this theme involving categories with coproducts,
or finite colimits, or finite coproducts? 

I can think of three things of mine in print which are relevant:

\item"{[35]}" (with F. Foltz and C. Lair) Algebraic categories with few
monoidal biclosed structures or none, {\it Jour. Pure and Applied Alg.} 17
(1980), 171-177.

(the last section or so of)
\item"{[41]}" Structures defined by finite limits in the enriched context I,
 {\it Cahiers de Top. et G\'eom. Diff.} 23 (1982), 3-42.

\item"{[52]}" (with G.B. Im) A universal property of the convolution monoidal
structure, {\it J. Pure Appl. Algebra} 43 (1986), 75-88. 

Much of this kind of thing is folklore; when one uses "left adjoint" rather
than "cocontinuous", some speak of considering "objects in two categories" -
see the first paper above.

Max Kelly.