By Jiri Adamek
This up to date introductory remedy employs classification concept to discover the speculation of constructions. Its new angle stresses concrete different types and provides a scientific view of factorization constructions, providing a unifying standpoint on previous paintings and summarizing contemporary advancements. a number of examples, starting from normal to precise, remove darkness from the textual content. 1990 version, up to date 2004.
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Additional info for Abstract and concrete categories: the joy of cats
Proof: (1). One direction is immediate and the other is a consequence of the fact that (full) embeddings are closed under composition. (2). One direction follows from the compositive nature of faithful functors. To show, conversely, that every faithful functor can be decomposed as stated, let E1 : D → B be the inclusion of the full subcategory D of B that has as objects all images (under F ) of A-objects. Let C be the category with Ob(C) = Ob(A), with homC (A, A ) = homB (F A, F A ), and with identities and composition defined as in B.
1) If F and G are both isomorphisms (resp. embeddings, faithful, or full), then so is G ◦ F. (2) If G ◦ F is an embedding (resp. faithful), then so is F . (3) If F is surjective on objects and G ◦ F is full, then G is full. 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.
However, if A is the monoid of all maps from into , considered as a category, then the subcategory of A, consisting of all maps f : → with f (0) = 0, is reflective in A. ◆ ◆ ◆ ◆ (f) Show that the (nonfull) subcategory A of Pos, consisting of those posets for which every nonempty subset has a meet, and those morphisms that preserve meets of id nonempty subsets, is reflective in Pos. For a poset A, A −−→ A is an A-reflection if and only if A is well-ordered. (g) Verify that the category • GG • has no proper reflective subcategory.
Abstract and concrete categories: the joy of cats by Jiri Adamek