David Leduc <david.leduc6@googlemail.com> asked
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)? ... [remainder snipped] ...
The key here is to understand what to mean by "in a similar manner," and for that one needs to understand what the "manner" is by which one comes up with the diagonal morphisms themselves originally. As I see it, the original "manner" for the diagonal morphism is this: in the category of Sets, the only function available from 2 to 1 is the universal example (in contravariant form) of such a diagonal morphism -- 1) in Sets^op, 2 is the product of two copies of 1, and the function just mentioned is that diagonal morphism of yours for that product; 2) anywhere at all, your "diagonal morphism" C --> C*C is just what that function from 2 to 1 induces as morphism from C^1 (which "is" C) to C^2 (which "is" your C*C). So, now, by "similar manner" I would have to understand some sort of family of maps C*C --> C or C^2 --> C^1 induced by some 1 --> 2. Well, there are only two functions 1 --> 2 (the two injections), and neither of them does what you'd like (not even in any particular special case of C). What this proves, beyond any shadow of a doubt, is that, if there *is* any sense of "similar manner" for which your question has an affirmative answer, it sure as heck isn't what "I would have to understand," as just described. What's worse, I wouldn't expect to achieve *any* uniform ways of coming up with maps C*C --> C that deliver binary products in the case of a category C that *has* products -- what, after all, could it be delivering for a category C that just happens *not* to be "closed under products" :-) ? Does this help you at all? Cheers -- Fred
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!
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