Naive category theory

Abstract

Our objective in this chapter is to assemble some basics of the theory and to acquire some practice with the language of categories. There are several very good introductory texts for mathematicians and the ones by MacLane [67] and Herrlich and Strecker [46] are probably the best references for physicists who want more details than we can provide here. Note that we use the term set where MacLane uses the term small set: to mean a class that is not a proper class ( cf . [46]). The important te chni cal point is that we must not speak of the ‘set of all sets’ or ‘the set of all groups’ these are not constructible from the operations of standard set theory; they are examples of proper classes. We shall suppose that any group, ring, field, vector space or topological space is a set together with some extra structure, that is a class which is not a proper class. Occasionally we shall need to use maps between classes, then we use the same notation as for maps between sets.