Bicategorical Fibration Structures and Stacks

In this paper we introduce two notions —systems of fibrant objects and fibration structures— 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 add pseudo-inverses to weak equivalences in B. Furthermore, Ho(B) is locally small when B is and Ho(B) is a 2-category when B is. We thereby resolve two of the problems with known approaches to bicategorical localization. As an important example, we describe a fibration structure on the 2-category of prestacks on a site and prove that the resulting homotopy bicategory is the 2-category of stacks. We also show how this example can be restricted to obtain algebraic, differentiable and topological (respectively) stacks as homotopy categories of algebraic, differential and topological (respectively) prestacks. Introduction It is widely known that Quillen’s [19] notion of model structure on a category C provides a technical tool for forming the localization of C with respect to a class of weak equivalences: weak equivalences are inverted in a universal way in the passage to the homotopy category Ho(C) of C. Consequently, it is possible to invert weak equivalences in this setting without having to resort to the Gabriel-Zisman [7] calculus of fractions. One advantage of using model structures for localization is that the resulting homotopy category will be locally small when C is. This is not necessarily the case for the calculus of fractions. In the bicategorical setting, one might like to be able to invert a collection of weak equivalences in the sense of turning them into equivalences. In [17, 18], the first author gave a bicategorical generalization of the Gabriel-Zisman calculus of fractions which accomplishes this goal: 0.1. Theorem. [Pronk [18]] Given a collection of arrows W in a bicategory C satisfying certain conditions, there exists an explicitly constructed bicategory C(W−1) (called the bicategory of fractions for W) and a homomorphism I : C → C(W−1) such that I sends arrows in W to equivalences in C(W−1) and I is universal with this property. Like the ordinary category of fractions, this construction suffers from the technical defect that C(W−1) will not in general have small hom-categories even when C does. Received by the editors 2014-05-07 and, in revised form, 2014-12-02. Transmitted by Tom Leinster. Published on 2014-12-08. 2010 Mathematics Subject Classification: Primary: 18D05; Secondary: 18G55, 14A20.