# A strange ring category

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.

• 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

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 assume that you mean a group object in the nonstandard category of algebras where also anti-homomorphisms are allowed? But this probably has no products, and Hopf algebras are far from being group objects, since the multiplication is not defined on a product. Cocommutative Hopf algebras are group objects in the category of cocommutative coalgebras (here $\otimes=\times$). – Martin Brandenburg Jun 6 '15 at 20:48
• @MartinBrandenburg You are right. I can save my answer some by pointing out that I was right that antipode is a map of rings in this peculiar category, but you are right that $\otimes$ is not product here. – David E Speyer Jun 7 '15 at 0:09
Let $\mathcal{C}$ be a category. Let $D : \mathcal{C} \to \mathcal{C}$ be a functor with $D \circ D = \mathrm{id}$. For $\mathcal{C}=\mathsf{Mon},\,\mathsf{Grp},\,\mathsf{Ring}$ or $\mathsf{Cat}$ we take $D$ to be the "opposite-object" functor. Define a new category $\mathcal{C}_D$ as follows: The objects are the objects of $\mathcal{C}$. For objects $x,y$ we define $$\hom_{\mathcal{C}_D}(x,y) = \hom_{\mathcal{C}}(x,y) \sqcup \hom_{\mathcal{C}}(x,D(y)).$$ The identity of an object in $\mathcal{C}_D$ is the identity in $\mathcal{C}$. The composition of $x \to y$ with $y \to z$ in $\mathcal{C}_D$ is the one in $\mathcal{C}$. The composition of $x \to y$ with $y \to D(z)$ is the composition $x \to D(z)$ in $\mathcal{C}$, regarded as an element in the second part of $\hom_{\mathcal{C}_D}(x,z)$. The composition of $x \to D(y)$ with $y \to z$ is the composition $x \to D(y) \to D(z)$ in $\mathcal{C}$, again regarded as an element in the second part of $\hom_{\mathcal{C}_D}(x,z)$. Finally, the composition of $x \to D(y)$ with $y \to D(z)$ is the composition $x \to D(y) \to D(D(z)) = z$, now regarded as an element in the first part of $\hom_{\mathcal{C}_D}(x,z)$. One checks that $\mathcal{C}_D$ is, indeed, a category.