We introduce and describe the 2-category Grt♭ of
Grothendieck categories and flat morphisms between them. First, we show that
the tensor product of locally presentable linear categories ⊠
restricts nicely to Grt♭. Then, we characterize exponentiable
objects with respect to ⊠: these are 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.Comment: Minor revision. The proofs of Sec 5 have been expanded to make the
paper self containe