Homology, Homotopy and Applications

Volume 3 (2001)

Number 2

Volume of a Workshop at Stanford University

Stacks and the homotopy theory of simplicial sheaves

Pages: 361 – 384

DOI: https://dx.doi.org/10.4310/HHA.2001.v3.n2.a5


J. F. Jardine (Department of Mathematics, University of Western Ontario, London, Ontario, Canada)


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.

2010 Mathematics Subject Classification

Primary 18G50. Secondary 14A20, 18F20, 18G30.

Published 1 January 2001