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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Abstract homotopy theory and generalized sheaf cohomology

Author: Kenneth S. Brown
Journal: Trans. Amer. Math. Soc. 186 (1973), 419-458
MSC: Primary 55B30; Secondary 18H99
MathSciNet review: 0341469
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Abstract: Cohomology groups ${H^q}(X,E)$ are defined, where X is a topological space and E is a sheaf on X with values in Kan’s category of spectra. These groups generalize the ordinary cohomology groups of X with coefficients in an abelian sheaf, as well as the generalized cohomology of X in the usual sense. The groups are defined by means of the “homotopical algebra” of Quillen applied to suitable categories of sheaves. The study of the homotopy category of sheaves of spectra requires an abstract homotopy theory more general than Quillen’s, and this is developed in Part I of the paper. Finally, the basic cohomological properties are proved, including a spectral sequence which generalizes the Atiyah-Hirzebruch spectral sequence (in generalized cohomology theory) and the “local to global” spectral sequence (in sheaf cohomology theory).

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Keywords: Abstract homotopy theory, generalized sheaf cohomology, homotopical algebra, sheaf of spectra, homotopy category, derived functor
Article copyright: © Copyright 1973 American Mathematical Society