  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.
