By P. J. Hilton, U. Stammbach (auth.)
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Extra info for A Course in Homological Algebra
O is the category of spaces-with-base-point (see ). :~ -+ (fj, where the subscript h indicates that the morphisms are to be regarded as (based) homotopy classes of (based) continuous functions. :~ and then 7t factors as 7t = 1fQ. :h-+Illb). (g) We saw in Chapter I how the set 9Jl~(A, B) = HomA(A,B) may be given the structure of an abelian group. If we hold A fixed and define II. 21b by 9Jl~(A, - )(B) = 9Jl~(A, B), then 9Jl~(A, -) is a functor. More generally, for any category (£: and object A of (£:, (£:(A, -) is a functor from (£: to 6.
21 b). (c) Let A be an object of9Jl~ and let G be an abelian group. We saw in Section I. a left A-module. 47 2. Functors Hom z ( -, G) thus appears as a contravariant functor from 9Jl~ to 9Jl~. Further examples will appear as exercises. Finally we make the following definitions. Recall from Section 1 the notion of a full subcategory. Consistent with that definition, we now define a functor F : (t-:D as full if F maps (t(A, B) onto :D(F A, F B) for all objects A, B in (t, and as faithful if F maps (t(A, B) injectively to :D(F A, F B).
2. Let G be a divisible abelian group. Then X is an injective A-module. = Homz(A, G) Proof. Let J1: A-+B be a monomorphism of A-modules, and let a: A-+X a homomorphism of A-modules. We have to show that there exists P: B-+ X such that PJ1 = a. 1 to a homomorphism of abelian groups a' : A -+ G. Since G is injecti ve, there exists P' : B -+ G such that P' J1 = a'. 1 we obtain a homomorphism of A-modules p: B-+A. Finally by the naturality of 1], the diagram is commutative. 3. 3. Every A-module A is a sub module of a cofree, hence injective, A-module.