By P. J. Hilton, U. Stammbach (auth.)
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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). Finally F is a full embedding if F is full and faithful and one-to-one on objects.
A) (£:( - , B), for B an object in (£:, is a contravariant functor from (£: to 6. n~ respectively to III b. We say that these functors are represented by B. (b) The (singular) cohomology groups are contravariant functors '.! ---. h ---. 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.
Of course, this could be proved using the explicit construction of P tB Q, but we prefer to emphasize the universal property of the direct sum. Next assume that P tB Q is projective. Let e: B-C be a surjection and yp : P-+C a homomorphism. Choose YQ: Q---+C to be the zero map. We obtain y: PtBQ-+C such that yip = yp and YIQ = YQ =0. Since PtBQ is projective there exists p: P$Q-+B such that ep = y. Finally we obtain e(f3lp) = yip = yp. Hence f3lp: P-+B is the desired homomorphism. Thus P is projective; similarly Q is projective.