Abstract Homotopy Theory and Generalized Sheaf Cohomology

HOMOTOPY THEORY AND GENERALIZED SHEAF COHOMOLOGY BY KENNETHS. BROWN0) ABSTRACT. Cohomology groups Ha(X, E) are defined, where X is a topological space and £ 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). Cohomology groups Ha(X, E) are defined, where X is a topological space and £ 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). Introduction. In this paper we will study the homotopy theory of sheaves of simplicial sets and sheaves of spectra. This homotopy theory will be used to give a derived functor definition of generalized sheaf cohomology groups H^iX, E), where X is a topological space and E is a sheaf of spectra on X, subject to certain finiteness conditions. These groups include as special cases the usual generalized cohomology of X defined by a spectrum [22] and the cohomology of X with coefficients in a complex of abelian sheaves. The cohomology groups have all the properties one would expect, the most important one being a spectral sequence Ep2q = HpiX, n_ E) =■» Hp+*iX, E), which generalizes the Atiyah-Hirzebruch spectral sequence (in generalized cohomology) and the "local to global" spectral sequence (in sheaf cohomology), and Received by the editors June 8, 1972 and, in revised form, October 15, 1972. AMS (MOS) subject classifications (1970). Primary 55B20, 55B30, 55D99, 18F99; Secondary 55J10.