Feeds:
Posts

Morita equivalence and the bicategory of bimodules

In the previous post we learned that it is possible to recover the center $Z(R)$ of a ring $R$ from its category $R\text{-Mod}$ of left modules (as an $\text{Ab}$-enriched category). For commutative rings, this justifies the idea that it is sensible to study a ring by studying its modules (since the modules know everything about the ring).

For noncommutative rings, the situation is more interesting. Two rings $R, S$ are said to be Morita equivalent if the categories $R\text{-Mod}, S\text{-Mod}$ are equivalent as $\text{Ab}$-enriched categories. As it turns out, there exist examples of rings which are non-isomorphic but which are Morita equivalent, so Morita equivalence is a strictly coarser equivalence relation on rings than isomorphism. However, many important properties of a ring are invariant under Morita equivalence, and studying Morita equivalence offers an interesting perspective on rings on general.

Moreover, Morita equivalence can be thought of in the context of a fascinating larger structure, the bicategory of bimodules, which we briefly describe.

The center $Z(G)$ of a group is an interesting construction: it associates to every group $G$ an abelian group $Z(G)$ in what is certainly a canonical way, but not a functorial way: that is, it doesn’t extend (at least in any obvious way) to a functor $\text{Grp} \to \text{Ab}$ (unlike the abelianization $G/[G, G]$). We might wonder, then, exactly what kind of construction the center is.
Of course, it is actually not hard to come up with a rather general example of a canonical but not functorial construction: in any category $C$ we may associate to an object $c \in C$ its automorphism group $\text{Aut}(c)$ or endomorphism monoid $\text{End}(c)$), and this is a canonical construction which again doesn’t extend in an obvious way to a functor $C \to \text{Grp}$ or $C \to \text{Mon}$. (It merely reflects some special part of the bifunctor $\text{Hom}(-, -)$.)
It turns out that the center can actually be thought of in terms of automorphisms (or endomorphisms), not of a group $G$, but of the identity functor $G \to G$, where $G$ is regarded as a category with one object. This definition generalizes, and the resulting general definition has some interesting specializations. Moreover, an important general property is that the center is always abelian, and this has a very elegant proof.