## Algebraic Topology: Lecture 12

We discussed adjunctions between two categories. If C and D are
categories, and adjunction from C to D is a triple (F,G,a) that
consists of a functor F : C --> D, a functor G : D --> C, and a
natural isomorphism
~
a : Hom_D(F(X),Y) ----> Hom_C(X,G(Y)).

We say that F is the left adjoint functor and that G is the right
adjoint functor. We give two examples. Firstly, let C be the category of
sets and maps, and let D be the category of abelian groups and group
homomorphisms. Let F be the functor that to a set X associates the
free abelian group F(X), and let G be the functor that to an abelian
group A associates the underlying set G(A) of A. We say that G is a
forgetful functor because it forgets the abelian group structure on
the set A. Then there is an adjunction (F,G,a), where the adjunction
isomorphism a is given by a(f)(x) = f(x). Secondly, let I be a small
category (this means that the class of objects forms a set), and let C
be any category. Let us assume that colimits and limits indexed by I
exist in C. We let C^I be the category of I-diagrams in C. It is
defined by
ob(C^I) = all functors X : I --> C.
mor(C^I) = all natural transformations between such functors.

We define the diagonal functor
diag : C ---> C^I

by C(X)(i) = X, C(X)(i -> i') = id_X, and C(f)_i = f. Then there is
an adjunction (colim_I,diag,a) where the isomorphism
a : Hom_C(colim_I X, Y) ---> Hom_{C^I}(X,diag(Y))

takes the morphism f on the left-hand side to the natural
transformation a(f) on the right-hand side whose value at the objects
i in I is given by the composite map
g_i f
a(f)_i : X_i ------> colim_I X ----> Y

where g_i is the canonical map. Similarly, there is an adjunction
(diag,lim_I,a). This just amounts to a reformulation of the definition
of the colimit and limit of a diagram.
Suppose that (F,G,a) is an adjunction between two model categories C
and D. We say that the adjunction is a Quillen adjunction and that F
(resp. G) is a left (resp. right) Quillen functor if F preserves
cofibrations and trivial cofibrations, or equivalently, if G preserves
fibrations and trivial fibrations. It turns out that a left Quillen
functor preserves all weak equivalences between cofibrant objects and
that a right Quillen functor preserves all weak equivalences between
fibrant objects. We defined the total left derived
functor LF : Ho(C) --> Ho(D). The total right derived
functor RG : Ho(D) --> Ho(C) is defined similarly. Moreover,
the Quillen adjunction (F,G,a) gives rise to an adjunction
(LF,RG,Ra). We will show this next time.