Question

This can be seen as a follow up my question here:

http://mathoverflow.net/questions/28152/is-there-a-notion-of-fibered-category-with-boxproducts

Given a monoidal fibration $f:E\rightarrow B$ (i.e. a strict monoidal functor between monoidal categories which is a fibration of ordianary categories) where the base is a cartesian monoidal category endowed with a grothendieck topology. What are the right conditions for such a fibration be called a stack?

I guess it is not enough to ask that $E(X)\rightarrow Desc(X,U)$ is an equivalence of ordinary categories. Insted one should need some further condition that ensures the following:

"if $(\phi_i)$ can be glued to $\phi$ and $(\psi_j)$ can be glued to $\psi$ than $(\phi_i\boxtimes \psi_j)$ can be glued to $\phi\boxtimes \psi$"

Does this notion exist yet? What would be the right condition?

Examples I have in mind are

$B$=geometric objects for example smooth varieties and

$E$=sheaves for example $\mathcal{D}_X$-modules

Answer 1

I would take the view point that a monoidal category is a bicategory with one object. (Then a category fibered in monoidal categories should be the same thing as a weak functor into bicategories that "only hits monoidal categories".) In other words, what you should have is that this fibration is a 2-stack when viewed as a fibration in bicategories. The descent condition should then be that the canonical map $E|_X \to Desc(X,U)$ be an equivalence of bicategories, where each of these monoidal categories is viewed as a bicategory, which is equivalent to Jeff's comment; this is just saying that $E|_X \to Desc(X,U)$ is a monoidal equivalence.