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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.

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I'm confused by your "product morphism": that would rather be a coproduct, no? The coproduct of rings is already nasty in the ordinary category of rings. –  t.b. Sep 11 '12 at 16:40
I think he just means composition of maps, not "product" in the sense of category theory, @t.b. –  Thomas Andrews Sep 11 '12 at 16:51
@ThomasAndrews No... it's pretty clear there's no issue with compositon of morphisms... –  rschwieb Sep 11 '12 at 17:04
@t.b. I think you're right... I always forget which one is the messy one. –  rschwieb Sep 11 '12 at 17:05
It doesn't help that I somehow left out a critical negating word... –  rschwieb Sep 11 '12 at 17:12

1 Answer 1

up vote 4 down vote accepted

This is useful is you would like a Hopf algebra to be a group object in the category of algebras. If $A$ is a Hopf algebra, then the antipode map $S: A \to A$, is an antiendomorphism of $A$. You can see this from the example of group algebras: if $G$ is a group, and $k[G]$ is the group algebra, then $k[G]$ has a Hopf algebra structure where $S: g \mapsto g^{-1}$ and we have $(gh)^{-1} = h^{-1} g^{-1}$.

I don't know that it's worth defining a category just to fix this issue. I have seen people spend far more time bothered by this issue than it is worth when learning Hopf algebras, so maybe this would have helped them.

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Oo, good lead... thank you. –  rschwieb Sep 11 '12 at 18:19

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