Recently I ran across a weird example of a category in Jacobson's Basic Algebra II.
The category has, as objects, the class of rings. As morphisms, it uses all ring homomorphisms and antihomomorphisms of these rings.
Has anyone seen a use for this category?
I have the sense that it isn't well behaved, and so it might only be useful as a counterexample.
For example, it seems like products don't work. I didn't verify any details, but if you suppose there are three noncommutative rings $R$ and $S$ and $T$ for which there is a homomorphism of $R$ into $T$ and an anti homomorphism of $S$ into $T$, it seems like a product morphism from "$R\oplus S$" to $T$ is unlikely to exist in general.
Of course, I may just be blinded by familiarity with nice categories, so maybe there is a way around it...
Added I may in fact mean the coproduct and not the product. I never remember which is the messy one, for rings. Anyhow, the idea is that if you use the normal Cartesian product with coordinatewise ring product, it doesn't seem possible for a single product/coproduct morphism to combine a homomorphism with an antihomomorphism.