Let be a commutative ring. A popular thing to do on this blog is to think about the Morita 2-category of algebras, bimodules, and bimodule homomorphisms over , but it might be unclear exactly what we’re doing when we do this. What are we studying when we study the Morita 2-category?

The answer is that we can think of the Morita 2-category as a 2-category of **module categories** over the symmetric monoidal category of -modules, equipped with the usual tensor product over . By the Eilenberg-Watts theorem, the Morita 2-category is equivalently the 2-category whose

- objects are the categories , where is a -algebra,
- morphisms are cocontinuous -linear functors , and
- 2-morphisms are natural transformations.

An equivalent way to describe the morphisms is that they are “-linear” in that they respect the natural action of on given by

.

This action comes from taking the adjoint of the enrichment of over , which gives a tensoring of over . Since the two are related by an adjunction in this way, a functor respects one iff it respects the other.

So Morita theory can be thought of as a categorified version of module theory, where we study modules over instead of over . In the simplest cases, we can think of Morita theory as a categorified version of linear algebra, and in this post we’ll flesh out this analogy further.