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Journal Article
Abstract Homotopy Theory and Generalized Sheaf Cohomology
Kenneth S. Brown
Transactions of the American Mathematical Society
Vol. 186 (Dec., 1973), pp. 419-458
Published
by: American Mathematical Society
DOI: 10.2307/1996573
https://www.jstor.org/stable/1996573
Page Count: 40
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Topics: Functors, Maps, Homotopy theory, Adjoints, Topological spaces, Factorization, Axioms, Spectral theory, Commutativity, Integers
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Abstract
Cohomology groups Hq(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).
Transactions of the American Mathematical Society
© 1973 American Mathematical Society