Homology, Homotopy and Applications

Stacks and the homotopy theory of simplicial sheaves

J. F. Jardine

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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

Permanent link to this document
http://projecteuclid.org/euclid.hha/1139840259

Mathematical Reviews number (MathSciNet)
MR1856032

Zentralblatt MATH identifier
0995.18006

Subjects
Primary: 18G50: Nonabelian homological algebra
Secondary: 14A20: Generalizations (algebraic spaces, stacks) 18F20: Presheaves and sheaves [See also 14F05, 32C35, 32L10, 54B40, 55N30] 18G30: Simplicial sets, simplicial objects (in a category) [See also 55U10]

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.


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