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/ ]
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/ ]
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/ ]
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!
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On 1/21/12, Fred E.J. Linton <fejlinton@usa.net> wrote:
Does this help you at all? Cheers -- Fred
This is very helpful to improve my understanding, thank you. However the reply by Todd was the one I was longing for: On 1/21/12, Todd Trimble <trimble1@optonline.net> wrote:
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.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (4)
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David Leduc -
Fred E.J. Linton -
Mike Stay -
Todd Trimble