Exponentiable Grothendieck categories in flat algebraic geometry

We introduce and describe the 2-category of Grothendieck categories and flat morphisms between them. First, we show that the tensor product of locally presentable linear categories ⊠ restricts nicely to Gr_flat. Then, we characterize exponentiable objects with respect to ⊠: these are the continuous Grothendieck categories. In particular, locally finitely presentable Grothendieck categories are exponentiable. Consequently, we have that, for a quasi-compact quasi-separated scheme X, the category of quasi-coherent sheaves QCoh(X) is exponentiable. Finally, we provide a family of examples and concrete computations of exponentials