Notion of stack fibered in monoidal categories? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T20:52:19Z http://mathoverflow.net/feeds/question/31173 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/31173/notion-of-stack-fibered-in-monoidal-categories Notion of stack fibered in monoidal categories? Jan Weidner 2010-07-09T11:44:28Z 2010-07-09T16:52:12Z <p>This can be seen as a follow up my question here:</p> <p><a href="http://mathoverflow.net/questions/28152/is-there-a-notion-of-fibered-category-with-boxproducts" rel="nofollow">http://mathoverflow.net/questions/28152/is-there-a-notion-of-fibered-category-with-boxproducts</a></p> <p>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?</p> <p>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:</p> <p>"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 $\phi\boxtimes \psi$"</p> <p>Does this notion exist yet? What would be the right condition? </p> <p>Examples I have in mind are </p> <p>$B$=geometric objects for example smooth varieties and</p> <p>$E$=sheaves for example $\mathcal{D}_X$-modules</p> http://mathoverflow.net/questions/31173/notion-of-stack-fibered-in-monoidal-categories/31206#31206 Answer by David Carchedi for Notion of stack fibered in monoidal categories? David Carchedi 2010-07-09T15:46:23Z 2010-07-09T15:46:23Z <p>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.</p>