Let $k$ be a field, and let $\alpha \in \mathrm{Br}(k)$. Let $A$ be an Azumaya algebra representing $\alpha$. Then the category $A$–$\mathrm{mod}$ depends only on $\alpha$.

I would like to know whether there's a way to describe which $k$-linear categories arise this way. Thus I'd like to know if there's a way to define the Brauer group of a field $k$ as classifying certain kinds of $k$-linear categories. I'd also like to know if there's a good description of the sum of elements of the Brauer group in terms of categories (is it some sort of tensor product of categories?).

The one condition I can come up with is that it should be a semisimple abelian category over $k$ for which the endomorphism algebra of the unit object is $k$.

An even more bold hope is to express the invariant map $\mathrm{Br}(\mathbb{Q}_p) \to \mathbb{Q}/\mathbb{Z}$ in terms of this category.