Higher linear algebra

Annoying Precision

Let $latex k$ be a commutative ring. A popular thing to do on this blog is to think about the Morita 2-category $latex text{Mor}(k)$ of algebras, bimodules, and bimodule homomorphisms over $latex k$, 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 $latex text{Mod}(k)$ of $latex k$-modules, equipped with the usual tensor product $latex otimes_k$ over $latex k$. By the Eilenberg-Watts theorem, the Morita 2-category is equivalently the 2-category whose

  • objects are the categories $latex text{Mod}(A)$, where $latex A$ is a $latex k$-algebra,
  • morphisms are cocontinuous $latex k$-linear functors $latex text{Mod}(A) to text{Mod}(B)$, and
  • 2-morphisms are natural transformations.

An equivalent way to describe the morphisms is that they are « $latex text{Mod}(k)$-linear » in that they…

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