Generalities on bundles and categories

Abstract

This is a preliminary chapter introducing much of the general terminology in the topology of bundles with the general language of category theory. We use the term bundle in the most general context, and then in the next chapters, we define the main concepts of our study, that is, vector bundles, principal bundles, and fibre bundles, as bundles with additional structure. In Grothendieck's notes "A General Theory of Fibre Spaces with Structure Sheaf," University of Kansas, Lawrence, 1955 (resp. 1958) he mentions that "the functor aspect of the notions dealt with has been stressed through, and as it now appears should have been stressed even more." Hopefully, we are carrying this out to the appropriate extent in this approach to bundles mixed with a general introduction to category theory where examples are drawn from the theory of bundles. The reader with a background in category theory and topology will see only a slightly different approach from the usual one. We will introduce several notations used through the book, for example, (set), (top), (gr), and (k) denote, respectively, the categories of sets, spaces, groups, and k-modules for a commutative ring k. These are explained in the context of the definition of a category in Sect. 4 and in the notations for categories at the end of the book. Chap. 2 of Fibre Bundles (Husemöller 1994) is a reference for this chapter. © Springer-Verlag Berlin Heidelberg 2008.