Title: Abstract homotopy theory and generalized sheaf cohomology
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 Kanâs category of spectra. These groups generalize the ordinary cohomology groups of X...Cohomology groups ${H^q}(X,E)$ are defined, where X is a topological space and E is a sheaf on X with values in Kanâs 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 âhomotopical algebraâ 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 Quillenâs, 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 âlocal to globalâ spectral sequence (in sheaf cohomology theory).Read More