The aim of this chapter is to fix some notation and recall well-known facts concerning basic concepts of category theory used throughout the book. The reader may well skip it and return to it when needed. Only the most usual definitions and results of the theory of categories are mentioned here; more about them can be found in any of the books mentioned at the end of this chapter.

Foundations In category theory, one needs to distinguish between small collections (sets) and large ones (classes). An arbitrary set theory making such a distinction possible is sufficient for our book. The category of (small) sets and functions is denoted by

Set.

All categories with which we work have small hom-sets. It follows that every object has only a set of retracts (see 0.16) up to isomorphism.

Properties of functors A functor F: A → B is

1. faithful if for every parallel pair of morphisms f, g: A ⇉ A′ in A, one has f = g whenever Ff = Fg

2. full if for every morphism b: FA → FA′ in B, there exists a morphism a: A → A′ in A such that Fa = b

3. essentially surjective if for every object B in B, there exists an object A in A with B isomorphic to FA

4. […]