tensor product of categories

tensor product of categories

[john baez]

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?  

Re: tensor product of categories

[Max Kelly]

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.