Appendix A - Some points of category theory

Summary

In this book, category theory is used extensively, if at an elementary level. The notions of category, functor and natural transformation are used throughout, together with standard tools like limits, colimits and adjoint functors.

The brief review of this appendix is also meant to fix the notation used here. Proofs can be found in the texts mentioned in Section A1.1, except for some non-standard points at the end of this chapter.

Basic notions

Smallness

Something must be said about set-theoretical aspects, to make precise the meaning of the category of ‘all' sets’, or ‘all’ topological spaces, and so on.

We work within the NBG (von Neumann–Bernays–Göodel) theory, where there are sets and classes, and the class of all sets or all spaces makes sense. In a category A the objects form a class ObA and the morphisms form a class MorA; but, for every pair X, Y of objects, we assume that the morphisms X → Y, also called maps or arrows, form a setA(X, Y). The category is said to be small if the class ObA is a set, and large otherwise.