## Homology, Homotopy and Applications

### Stacks and the homotopy theory of simplicial sheaves

J. F. Jardine

#### Abstract

Stacks are described as sheaves of groupoids $G$ satisfying an effective descent condition, or equivalently such that the classifying object $BG$ satisfies descent. The set of simplicial sheaf homotopy classes $[*,BG]$ is identified with equivalence classes of acyclic homotopy colimits fibred over $BG$, generalizing the classical relation between torsors and non-abelian cohomology. Group actions give rise to quotient stacks, which appear as parameter spaces for the separable transfer construction in special cases.

#### Article information

Source
Homology Homotopy Appl. Volume 3, Number 2 (2001), 361-384.

Dates
First available in Project Euclid: 13 February 2006

http://projecteuclid.org/euclid.hha/1139840259

Mathematical Reviews number (MathSciNet)
MR1856032

Zentralblatt MATH identifier
0995.18006

#### Citation

Jardine, J. F. Stacks and the homotopy theory of simplicial sheaves. Homology Homotopy Appl. 3 (2001), no. 2, 361--384. http://projecteuclid.org/euclid.hha/1139840259.