By Jiri Adamek

ISBN-10: 0471609226

ISBN-13: 9780471609223

This up to date introductory remedy employs classification thought to discover the speculation of constructions. Its new angle stresses concrete different types and provides a scientific view of factorization buildings, delivering a unifying point of view on previous paintings and summarizing fresh advancements. quite a few examples, starting from normal to precise, light up the textual content. 1990 version, up-to-date 2004.

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Functors (a) Consider the category A with exactly one object A, with hom(A, A) = {a, b}, and with composition defined by a ◦ a = a and a ◦ b = b ◦ a = b ◦ b = b. Consider further the category B with exactly one object B, with hom(B, B) = {b}, and with b ◦ b = b. Let F be defined by F (B) = A, F (b) = b. Is F : B → A a functor? (b) Show that a category A is thin (resp. empty) if and only if every functor with domain A is faithful (resp. full). 3H. Equivalences (a) Show that none of the following categories is equivalent to any of the others: Set, Vec, Grp, Top, Rel, Pos.

31 PROPOSITION If F : A → B is a full, faithful functor, then for every B-morphism f : F A → F A , there exists a unique A-morphism g : A → A with F g = f . Furthermore, g is an A-isomorphism if and only if f is a B-isomorphism. Proof: The morphism exists by fullness, and it is unique by faithfulness. 21 functors preserve isomorphisms, f is an isomorphism if g is. If f : F A → F A is a B-isomorphism, let g : A → A be the unique A-morphism with F (g ) = f −1 . Then F (g ◦ g) = F g ◦ F g = f −1 ◦ f = idF A = F (idA ), so that by faithfulness g ◦ g = idA .

17 Then: (a) R-Mod is isomorphic to Mod-R, for any ring R. ❩ ❩ (b) If denotes the ring of integers, then -Mod is isomorphic to the construct Ab of abelian groups and group homomorphisms. 20(13)] and action homomorphisms [f (m ∗ x) = m ∗ f (x)]. 3(2)(e)]. 27 DEFINITION Let F : A → B be a functor. (1) F is called an embedding provided that F is injective on morphisms. (2) F is called faithful provided that all the hom-set restrictions F : homA (A, A ) → homB (F A, F A ) are injective. (3) F is called full provided that all hom-set restrictions are surjective.

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