Chapter 0

PRELIMINARIES

To begin with, we state all basic definitions and theorems which we

refer to in the sequel, but with which a non-specialist is not likely to

be familiar. This does not make this work self-contained, for we will not

give their proofs. Rather, it is intended as a guide to detailed refer-

ences, which will be given for each theorem mentioned in this chapter, and

which will be kept visible in the sequel. Thus, the best way to use this

chapter is to consider it as a quick reference: not to read it,at the

beginning, but refer to it as the need arises, and then possibly consult

other references.

0.1. Adjoint Functors. Equivalence of Categories.

F

Let (E,X be two categories, ( E ^LZ^^ two functors. We say that

G

F is a left adjoint of G (i.e. G is a right adjoint of F ) if there is

a 1-1 correspondence:

F(X) • » Y in (T,

X - * G(Y) in D

natural in X and Y . We write F H G , and say that (F,G) is an

adjoint pair of functors. It is called an equivalence of categories if

FG is naturally isomorphic to L and GF is naturally isomorphic to

1_ . Categories ( E and X are said to be equivalent if there is an

equivalence of categories (F,G) between them.

Left adjoints preserve colimits, right adjoints preserve limits,

(cf. [Fre0] ,[ML]).

0.2. Presheaf Categories. Yoneda Lemma.

Let S be the category of sets and mappings. For a small category

( C , we write S for the category of presheaves on C , i.e. contra-

(Eop

variant functors from ( C to S . S has all (small) limits and co-

Received by the editors September 11, 1981, and in revised form May 3, 1983.

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