Sheaves and Presheaves

Abstract

In this chapter we shall develop the basic properties of sheaves and presheaves and shall give many of the fundamental definitions to be used throughout the book. In Sections 2 and 5 various algebraic operations on sheaves are introduced. If we are given a map between two topological spaces, then a sheaf on either space induces, in a natural way, a sheaf on the other space, and this is the topic of Section 3. Sheaves on a fixed space form a category whose morphisms are called homomorphisms. In Section 4, this fact is extended to the collection of sheaves on all topological spaces with morphisms now being maps f of spaces together with so-called f-cohomomorphisms of sheaves on these spaces. In Section 6 the basic notion of a family of supports is defined and a fundamental theorem is proved concerning the relationship between a certain type of presheaf and the cross-sections of the associated sheaf. This theorem is applied in Section 7 to show how, in certain circumstances, the classical singular, Alexander-Spanier, and de Rham cohomology theories can be described in terms of sheaves.