By Jiri Adamek
This updated introductory remedy employs type concept to discover the speculation of constructions. Its new angle stresses concrete different types and offers a scientific view of factorization buildings, supplying a unifying point of view on prior paintings and summarizing contemporary advancements. a number of examples, starting from basic to express, light up the textual content. 1990 variation, up to date 2004.
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The first fact follows from the commutativity of the diagram B idB rB B G RB rB idRB G RB and the second one from the commutativity of the diagram B g ◦f rB B G RB rB Rg ◦Rf G RB f g obtained by pasting together the corresponding diagrams for B −→ B and B −→ B . 18th January 2005 56 Categories, Functors, and Natural Transformations [Chap. 23 DEFINITION A functor R : B → A constructed according to the above proposition is called a reflector for A. 24 REMARK If A is a reflective subcategory of B, then a reflector for A depends upon the choice of the reflection arrows.
However, if we restrict our attention to categories that are sets, then both problems are eliminated. 44 DEFINITION A category A is said to be small provided that its class of objects, Ob(A), is a set. Otherwise it is called large. 45 REMARK Notice that when Ob(A) is a set, then M or(A) must be a set, so that the category A = (Ob(A), hom, id, ◦) must also be a set (cf. Exercise 3M). 46 EXAMPLES Mat is small; so are all preordered sets considered as categories, and all monoids considered as categories.
G H (b) There exist functors A −−→ C and C −−→ B with the following properties: (1) F = H ◦ G, (2) G is bijective on objects, (3) H is full and faithful, F G H (4) whenever A −−→ B = A −−→ C −−→ B and H is full and faithful, then there K exists a unique functor C −−→ C such that the diagram (∗) above commutes. ). In this section we investigate this phenomenon more thoroughly. 1 DEFINITION (1) A category A is said to be a subcategory of a category B provided that the following conditions are satisfied: (a) Ob(A) ⊆ Ob(B), (b) for each A, A ∈ Ob(A), homA (A, A ) ⊆ homB (A, A ), (c) for each A-object A, the B-identity on A is the A-identity on A, (d) the composition law in A is the restriction of the composition law in B to the morphisms of A.
Abstract and concrete categories: the joy of cats by Jiri Adamek
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