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.