I think you're "level slipping". If C is a monoidal category, there's a functor from C x C to C taking (c, c) to (c tensor c). Cat is monoidal under the cartesian product, so there's a morphism from (Cat, Cat) to Cat taking (C, C) to C x C. Likewise in any other category with products. On Sat, Jan 21, 2012 at 1:22 AM, David Leduc <david.leduc6@googlemail.com> wrote:
Hi,
In any category with products, one can define the diagonal morphism (from an object X to the object X*X). In the special case of the category of small categories, the above definition gives rise to the diagonal functor (from a category C to the category C*C).
How can I in a similar manner define the product functor (from the category C*C to the category C)? I.e. in any category with products, is there a morphism from X*X to X that would give rise to the product functor in the special case of the category of small categories? I guess that I need to assume something on X, but I cannot find the right assumption.
Thanks for any help!
-- Mike Stay - metaweta@gmail.com http://www.cs.auckland.ac.nz/~mike http://reperiendi.wordpress.com [For admin and other information see: http://www.mta.ca/~cat-dist/ ]