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.
1
Previous Page Next Page