Homotopy-systems, H-spaces and sheaf cohomology

The concepts of sheaves and sheaf cohomology are central throughout the work. Certain natural generalizations of these concepts are investigated in the latter part of the dissertation. The induced sheaf of a locally constant sheaf under a homotopy of a map of base spaces is shown to behave similar to the induced bundle of a locally constant bundle space, with respect to a homotopy of a map of the base spaces. The question: Are all sheaves limits of locally constant sheaves? is answered in the negative by demonstrating that such sheaves inherit certain homotopy properties of locally constant sheaves. Several related sheaf cohomology mapping theorems are proved, using sheaf cohomology with coefficients in locally constant sheaves or restrictions on the mapping or both, thus giving results concerning sheaf cohomology and homotopy type. A continuity theorem for a system of locally constant sheaves over a homotopy-inverse system of spaces is proved. (Homotopy-systems of spaces are introduced and investigated in the beginning of the work and numerous applications are found throughout the dissertation.) By relaxing the algebraic structure on the stalks of a sheaf to admit H-structures, the concept of a sheaf of H-spaces is introduced. A cohomology theory with coefficients in a sheaf of H-spaces is defined using the Cech technique. This cohomology theory is shown to satisfy Cartan's axioms for a sheaf cohomology theory. Other properties are explored and the theory is shown to contain the sheaf cohomology theory.