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.
Keywords: stacks, fibrant objects, homotopy bicategory, bicategories of fractions, algebraic stacks, differentiable stacks, topological stacks
2010 MSC: Primary: 18D05; Secondary: 18G55, 14A20
Theory and Applications of Categories, Vol. 29, 2014, No. 29, pp 836-873.