A Course in Homological Algebra by P. J. Hilton, U. Stammbach (auth.)

By P. J. Hilton, U. Stammbach (auth.)

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Categories and Functors 44 subcategories, that is, those subcategories (to of (t such that (to (A, B) = (t(A, B) for any objects A, B of (to. For example, 'llb is a full subcategory of G>, but 91 1 is a subcategory of 91 which is not full. 1. Show how to represent an ordered set as a category. (Hint: Regard the elements a, b, ... ) Express in categorical language the fact that the ordered set is directed [16]. Show that a subset of an ordered set, with its natural ordering, is a full subcategory.

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. We say that this functor is represented by A. It is an important question whether a given functor (usually to 6) may be represented in this sense by an object of the category. In viewing the last example the reader will have noted an asymmetry.

B) = ~(a) + AC. We have to check that this definition is consistent. b E A, we have Ii (A b) = AC. But A = ~ Ao for some ~ E A and therefore Ab = ~ Ao b. Hence ~(Ab) =~(~Aob) = ~~(Aob) = ~AoC =AC. Since (A,~) < (A, Ii), this contradicts the maximality of (A, ~), so that A = B as desired. 2. Every quotient of a divisible module is divisible. Proof. Let e: D-- E be an epimorphism and let D be divisible. For e E E and 0=1= A E A there exists dE D with e(d) = e and d' E D with Ad' = d. Setting e' = e(d') we have Ae' = Ae(d') = e(Ad') = e(d) = e.

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