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Cocycle Categories

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Algebraic Topology

Part of the book series: Abel Symposia ((ABEL,volume 4))

Abstract

The cocycle category H(X; Y) is defined for objects X and Y in a model category, and it is shown that the set of homotopy category morphisms [X, Y] is isomorphic to the set of path components of H(X; Y ), provided that the ambient model category s is right proper, and if weak equivalences are closed under finite products. Various applications of this result are displayed, including the homotopy classification of torsors, abelian sheaf cohomology groups, group extensions and gerbes. The older classification results have simple new proofs involving canonically defined cocycles. Cocycle methods are also used to show that the algebraic K-theory presheaf of spaces is a simplicial stack associated to a suitably defined parabolic groupoid.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Western Ontario, ON, N6A 5B7, London, Canada

    J. F. Jardine

Authors
  1. J. F. Jardine
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Correspondence to J. F. Jardine .

Editor information

Editors and Affiliations

  1. Universitet NTNU, Norges Teknisk Naturvitenskap., Trondheim, 7491, Norway

    Nils Baas

  2. College of Letters, Arts & Sciences, University of Southern California, S. Vermont Ave. 3620, Los Angeles, 90089-2532, U.S.A.

    Eric M. Friedlander

  3. University of Oslo, Department of Mathematics, Oslo, 0316, Norway

    Björn Jahren

  4. University of Oslo, Department of Mathematics, Oslo, 0316, Norway

    Paul Arne Østvær

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Jardine, J.F. (2009). Cocycle Categories. In: Baas, N., Friedlander, E., Jahren, B., Østvær, P. (eds) Algebraic Topology. Abel Symposia, vol 4. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01200-6_8

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