Homology, Homotopy and Applications

Stacks and the homotopy theory of simplicial sheaves

J. F. Jardine

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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

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

First available in Project Euclid: 13 February 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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]


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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