10 - Finitely presented functors

Summary

A fresh entry-point to the book, this chapter begins with the generalisation of rings to small preadditive categories and of modules to functors on such categories. The basic results on additive functor categories are presented and the category of finitely presented functors is shown to be equivalent to the category of pp-pairs and pp-definable maps. This, in turn, is equivalent to the free abelian category. A good part of the “dictionary” between “pp” and functorial concepts is given.

Functor categories

The generalisation of rings to small preadditive categories and modules to functors on such categories is explained and illustrated in Section 10.1.1. Section 10.1.2 presents the Yoneda embedding of a category into a functor category via representable functors. The generating set of representable functors is discussed in Section 10.1.3.

Some short introductions to the key category of finitely presented functors can be found in, for instance, [69], [183], [323].

Functors and modules

This section contains basic definitions and examples and shows the pointwise nature of some concepts for functors.

We have seen a variety of functors – homological functors and functors defined by pp conditions – from categories of modules to the category, Ab, of abelian groups. We have also seen, 1.2.31, that pp functors commute with direct limits, so, since every module is a direct limit of finitely presented modules (E.1.20), these functors are determined by their action on the category of finitely presented modules.