In this chapter we first give an account of the elementary language of category theory, and show how this gives a unified way of looking at many of the ideas we have been considering. We are led to look for convenient properties of the categories of sheaves and of presheaves over a given topological space, and we find that they each have a list of such properties which are summarised in the definition of abelian category.

However, the construction of cokernels differs in the two categories; this expresses what is perhaps the basic question in sheaf theory: to what extent does a sheaf epimorphism (a map of sheaves which is ‘locally’ surjective) have surjective section maps? This is studied further when we consider cohomology (Chapter 5).

Lastly, we consider what happens in a change of base space by a continuous map. We find that there is a covariant (that is, going in the same direction as the map) method of changing the base space of presheaves, and a contraviant (opposite direction) construction which is a generalisation of sheaf if ication. These are connected by an adjointness relation, which may be interpreted as expressing their universal nature. In the case of an inclusion map of a locally closed subspace, we also consider the process of extension by zero.