Notation: $k$ - field. "$k$-category" = $k$-linear abelian category. $Vect\_k$ - the $k$-category of $k$-vector spaces. For a field extension $K/k$ and a $k$-category $\mathcal{A}$, denote by $\mathcal{A}\_K$ the $K$-category consisting of objects $M \in \mathcal{A}$ together with a $k$-algebra homomorphism $K \to End(M)$. 


I would like to say that the Brauer group $Br(K/k)$ classifies $k$-categories $\mathcal{A}$ such that $\mathcal{A}\_K \approx Vect\_K$.


My question is how to state it more precisely. For example, if I use cocomplete $k$-categories admitting a compact projective generator, then it becomes correct. Is it still correct if I say presentable $k$-categories? I.e., if a presentable $k$-category $\mathcal{A}$ satisfies $\mathcal{A}\_K \approx Vect\_K$, is it true that $\mathcal{A}$ admits a compact projective generator.


Or maybe there is a better way to think about this.


Thanks



Brauer group classifying some splitting categories
Everything you might dream about is true if you work with dg categories instead of abelian $1$-categories. This is proved in Toën's paper on derived Azumaya algebras. In particular, your question about local possession of a compact generator is dealt with and forms a crucial part of the proof. David Gepner and I proved similar results for stable $\infty$-categories in our paper on Brauer groups in derived algebraic geometry.


Less work has been done on the additive setting. Over fields, everything should be fine. But, over schemes, one runs into the problem of when $Br(X)=Br'(X)$ and the additional complication of non-torsion classes in $H^2\_{ét}(X,\mathbb{G}\_m)$. These classes classify étale twisted forms of the abelian category of quasi-coherent sheaves on X that globally do not admit compact projective generators (even when $X$ is affine).


