# Notion of stack fibered in monoidal categories?

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

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

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It looks like what you are asking for is that $E(X) \to Desc(X,U)$ is a monoidal equivalence. – Jeffrey Giansiracusa Jul 9 '10 at 11:48
Yes, I also thought about this, however I was confused and saw a problem which was no problem. Do you know whether there is a criterion when s monoidal functor is an monoidal equivalence? Is full, faithful and essential surjective enough? – Jan Weidner Jul 9 '10 at 19:14
@ Jan. Yes. If you have a monoidal functor, which is an equivalence of categories after forgetting the monoidal part, then it is in fact a monoidal equivalence. Hence fully-faithful and essentially surjective is enough. – Chris Schommer-Pries Jul 10 '10 at 14:29
Thanks, so, I am a bit confused. The functor $E(X)\rightarrow Desc(X,U)$ is monoidal, so no extra condition is needed, since it is automatically an monoidal equivalence whenever it is an equivalence? – Jan Weidner Jul 10 '10 at 18:05

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.
I am a bit worried about what $Desc(X,U)$ should mean here, since where are going to 2-stacks. You should define it to be $Hom(S_U,E)$, where the $Hom$ is maps of fibrations of bicategories, where $S_U$ is the sieve generated by the covering family $U$ – David Carchedi Jul 9 '10 at 15:49