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Bicategorical fibration structures and stacks

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Dorette A. Pronk and Michael A. Warren

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.

Published 2014-12-08.

http://www.tac.mta.ca/tac/volumes/29/29/29-29.pdf

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