Here is a question somebody asked today which used to be answered in an older version of the stacks project, but which got excised a while ago.<\/p>\n
The question is: How different are the notions of a stack in groupoids and a sheaf of groupoids?<\/p>\n
The answer is that there are 2 differences. The first is a minor one: Although every stack in groupoids is equivalent to a split category fibred in groupoids, it is not always isomorphic to one. Here a split category fibred in groupoids over a category is the category associated to a contravariant functor from the category into the category of groupoids. Of course such a functor is nothing else than a presheaf F of groupoids on the site.<\/p>\n
The second difference is more serious. Namely, when you say that F is a sheaf, then apart from the requirement that morphisms descend you are only requiring that descent data for objects are effective for a somewhat restrictive class of descent data. In fact you are only requiring that if x_i are objects of the split fibred category over the members U_i of the covering, and if the restrictions x_i|_{U_i \\times_U U_j} and x_j|_{U_i \\times_U U_j} are equal<\/strong> then this should be effective. Clearly this is different from the requirement that all descent data are effective.<\/p>\n The “explanation” of this in the earlier version of the stacks project is that the category F(U) should be the homotopy limit of the diagram<\/p>\n \\prod F(U_i) ==> \\prod F(U_i \\times_U U_j) ==> \\prod F(U_i \\times_U U_j \\times U_k) \u00a0…<\/p>\n and not the usual limit. And of course this is a nice way of saying it since it leads to possible generalizations such as higher stacks.<\/p>\n","protected":false},"excerpt":{"rendered":" Here is a question somebody asked today which used to be answered in an older version of the stacks project, but which got excised a while ago. The question is: How different are the notions of a stack in groupoids … Continue reading