Re: Existence of (co)product

kj wrote:
Does the product (or coproduct) always exist in a given category?
If not, is there a not-too-obscure example of a category for which
no such thing as a product or coproduct exists?

Take the category of fields (whose morphisms are ring homomorphisms;
i.e. inclusions, and zero maps). Then there is no product for _any_
two objects. Basically, this is because the Cartesian product of the
underlying sets, given the product ring structure, is never a field
(and always contains zero divisors). More precisely, let K and L be
fields and say their product, which I'll call P, exists. K and L have
a ring product K x L, and by the universal property for that we get a
map i : P -> L, which cannot be the zero map because then the
projections P -> K and P -> L would also be the zero maps, and they
have to factor into the identity maps K -> K and L -> L by the
universal property for P. Therefore i is injective, since P is a
field. Again since the projections P -> K and P -> L have to factor
into the identity maps, they must be surjective; therefore the image of
P within K x L surjects via its projection maps onto both K and L; it
follows by definition of K x L that in fact this image is all of it.
But of course K x L is not a field.

The coproduct of fields _may_ exist. For example, if the tensor
product of two fields, over the integers, is itself a field then by the
universal property of the tensor product of rings, and since the fields
are a full subcategory of the rings, it is the coproduct of fields as
well. For example, take the tensor product of the field with two
elements with itself: you get itself back. However, there can be no
coproduct for fields with different characteristics, since each factor
has to inject into the coproduct.

What can be said about the existence of the product (or coproduct)
of an arbitrary category? Is there a good way to characterize
those categories that have a product (or coproduct)?

This, I do not know. I do know that there is a characterization of
categories for which arbitrary limits (or colimits) exist, but it is
stated in terms of the existence of products (or coproducts)! Given
this I feel like there might not be a neat characterization for
products.

--
Ryan Reich
ryan.reich@xxxxxxxxx

.

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