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On the homotopy theory of sheaves of simplicial groupoids

  • André Joyal (a1) and Myles Tierney (a2)

The aim of this paper is to contribute to the foundations of homotopy theory for simplicial sheaves, as we believe this is the natural context for the development of non-abelian, as well as extraordinary, sheaf cohomology.

In [11] we began constructing a theory of classifying spaces for sheaves of simplicial groupoids, and that study is continued here. Such a theory is essential for the development of basic tools such as Postnikov systems, Atiyah-Hirzebruch spectral sequences, characteristic classes, and cohomology operations in extraordinary cohomology of sheaves. Thus, in some sense, we are continuing the program initiated by Illusie[7], Brown[2], and Brown and Gersten[3], though our basic homotopy theory of simplicial sheaves is different from theirs. In fact, the homotopy theory we use is the global one of [10]. As a result, there is some similarity between our theory and the theory of Jardine[8], which is also partially based on [10]

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[1] M. Barr . Toposes without points. J. Pure App. Alg. 5 (1974), 265280.

[2] K. S. Brown . Abstract homotopy theory and generalized sheaf cohomology. Trans. Amer. Math. Soc. 186 (1973), 419458.

[3] K. S. Brown and S. M. Gersten . Algebraic K-theory as generalized sheaf cohomology. Springer Lecture Notes in Mathematics 341 (1973), 266292.

[4] S. E. Crans . Quillen closed model structures for sheaves. J. Pure App. Alg. 101 (1995), 3557.

[7] L. Illusie . Complexe cotangent et déformations I & II, Springer Lecture Notes in Mathematics 239 and 283 (1971 & 1972).

[8] J. E. Jardine . Simplicial presheaves. J. Pure App. Alg. 47 (1987), 3587.

[9] J. E. Jardine . The homotopical foundations of algebraic K-theory. Contemporary Mathematics 83, (1989) 5782.

[11] A. Joyal and M. Tierney . Classifying spaces for sheaves of simplicial groupoids. J. Pure App. Alg. 89 (1993) 135161.

[12] A. Joyal and M. Tierney . Strong stacks and classifying spaces. Springer Lecture Notes in Mathematics 1488 (1991), 213236.

[14] D. M. Kan . On homotopy theory and c.s.s. groups. Annals of Mathematics 68 (1958), 3853.

[15] D. G. Quillen . Homotopical algebra. Springer Lecture Notes in Mathematics 43 (1967).

[16] D. G. Quillen . Rational homotopy theory. Annals of Math. 90 (1969), 205295.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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