This book is primarily concerned with the study of cohomology theories of general topological spaces with general coefficient systems. Sheaves play several roles in this study. For example, they provide a suitable notion of general coefficient systems. Moreover, they furnish us with a common method of defining various cohomology theories and of comparison between different cohomology theories.
The parts of the theory of sheaves covered here are those areas important to algebraic topology. Sheaf theory is also important in other fields of mathematics, notably algebraic geometry, but that is outside the scope of the present book. Thus a more descriptive title for this book might have been Algebraic Topology from the Point of View of Sheaf Theory.
Several innovations will be found in this book. Notably, the concept of the tautness of a subspace (an adaptation of an analogous notion of Spanier to sheaf-theoretic cohomology) is introduced and exploited throughout the book. The fact that sheaf-theoretic cohomology satisfies the homotopy property is proved for general topological spaces. Also, relative cohomology is introduced into sheaf theory. Concerning relative cohomology, it should be noted that sheaf-theoretic cohomology is usually considered as a single space theory. This is not without reason, since cohomology relative to a closed subspace can be obtained by taking coefficients in a certain type of sheaf, while that relative to an open subspace (or, more generally, to a taut subspace) can be obtained by taking cohomology with respect to a special family of supports. However, even in these cases, it is sometimes of notational advantage to have a relative cohomology theory. For example, in our treatment of characteristic classes in Chapter IV the use of relative cohomology enables us to develop the theory in full generality and with relatively simple notation. Our definition of relative cohomology in sheaf theory is the first fully satisfactory one to be given. It is of interest to note that, unlike absolute cohomology, the relative cohomology groups are not the derived functors of the relative cohomology group in degree zero (but they usually are so in most cases of interest).