Bicategorical fibration structures and stacks

Abstract

The familiar construction of categories of fractions, due to Gabriel and Zisman, allows one to invert a class W of arrows in a category in a universal way. Similarly, bicategories of fractions allow one to invert a collection of arrows in a bicategory. In this case the arrows are inverted in the sense that they are made into equivalences. As with categories of fractions, bicategories of fractions suffer from the defect that they need not be locally small even when the bicategory in which W lives is locally small. Similarly, in the case where W is a class of arrows in a 2-category, the bicategory of fractions will not in general be a 2-category. In this paper we introduce two notions ---systems of fibrant objects and fibration systems--- which will allow us to associate to a bicategory B a homotopy bicategory Ho(B) in such a way that Ho(B) is the universal way to invert weak equivalences in B. This construction resolves both of the difficulties with bicategories of fractions mentioned above. We also describe a fibration system on the 2-category of prestacks on a site and prove that the resulting homotopy bicategory is the 2-category of stacks. Further examples considered include algebraic, differentiable and topological stacks.