7 - Topological categories

Summary

Categorical methods have proved to be particularly useful when studying various questions in algebraic topology: today, most books on the subject contain a crash course in category theory, which turns out to provide a fruitful setting for handling the required structures. In this topic, the notions of abelian category and exact sequence play a key role.

This chapter is mainly concerned with the description of good categorical settings for developing general topology. And if this deserves a chapter in a book, this is clearly because the most obvious category one could think of – the category of topological spaces and continuous mappings – does not have rich categorical properties (like being regular, monadic, cartesian closed, a topos, …) which would have made applicable the results of some other chapters of this book.

In topology, one is mainly concerned with the problem of convergence and in particular problems of convergence in spaces of continuous functions. One is particularly interested in situations where “if a sequence of continuous functions converges, the limit is again continuous”. In fact, this requires a notion of convergence in the set of all functions (not necessarily continuous)… to express finally the continuity of the limit. Such a “good” situation is thus obtained when the set of continuous functions is closed in the set of all functions, for the corresponding topology inducing the notion of convergence.