Dear David, The only maps from X^2 to X that are guaranteed to exist in any category with products are the two projection maps. Indeed, if you consider the category opposite to finite sets, where the square of the one-element set 1 is the two-element set 2, there are precisely two maps 2 --> 1. (This is in some sense a universal example, as this category is equivalent to the initial Lawvere algebraic theory.) The question makes better sense in the context of 2-categories with 2-products, where a right adjoint to a diagonal map, if one exists, can rightfully be called a "product". This gives what you are after when applied to the 2-category Cat. One also finds this sort of thing in the context of cartesian bicategories, for example. Best wishes, Todd ----- Original Message ----- From: "David Leduc" <david.leduc6@googlemail.com> To: "categories" <categories@mta.ca> Sent: Saturday, January 21, 2012 4:22 AM Subject: categories: product functor, abstractly
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!
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]