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The Verdier Hypercovering Theorem

  Published:2011-05-17
 Printed: Jun 2012
  • J. F. Jardine,
    Mathematics Department, University of Western Ontario, London, ON   N6A 5B7
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Abstract

This note gives a simple cocycle-theoretic proof of the Verdier hypercovering theorem. This theorem approximates morphisms $[X,Y]$ in the homotopy category of simplicial sheaves or presheaves by simplicial homotopy classes of maps, in the case where $Y$ is locally fibrant. The statement proved in this paper is a generalization of the standard Verdier hypercovering result in that it is pointed (in a very broad sense) and there is no requirement for the source object $X$ to be locally fibrant.
Keywords: simplicial presheaf, hypercover, cocycle simplicial presheaf, hypercover, cocycle
MSC Classifications: 14F35, 18G30, 55U35 show english descriptions Homotopy theory; fundamental groups [See also 14H30]
Simplicial sets, simplicial objects (in a category) [See also 55U10]
Abstract and axiomatic homotopy theory
14F35 - Homotopy theory; fundamental groups [See also 14H30]
18G30 - Simplicial sets, simplicial objects (in a category) [See also 55U10]
55U35 - Abstract and axiomatic homotopy theory
 

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