A Homotopy Theory for Stacks

Sharon Hollander
Department of Mathematics, MIT
Cambridge, MA 02139
sharon@math.mit.edu

AMS Classification: Primary 14A20 ; Secondary 18G55, 55U10


We give a homotopy theoretic characterization of stacks on a site $\cC$
as the {\it homotopy sheaves} of groupoids on $\cC$.
We use this characterization to construct a model category
in which stacks are the fibrant objects.
We compare different definitions of stacks and show 
that they lead to Quillen equivalent model categories.
In addition, we show that these model structures 
are Quillen equivalent to the $S^2$-nullification 
of Jardine's model structure on sheaves of simplicial sets on $\cC$.