By Jiri Adamek

This updated introductory remedy employs type thought to discover the idea of constructions. Its special approach stresses concrete different types and provides a scientific view of factorization buildings, delivering a unifying viewpoint on previous paintings and summarizing contemporary advancements. various examples, starting from basic to express, light up the textual content. 1990 version, up-to-date 2004.

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54 shows that with every category we can associate an object-free category. Even though this correspondence is neither injective nor surjective, it provides an equivalence between the “standard” and the “object-free” definitions of category. This claim can be made precise as follows: (1) One can define functors between object-free categories to be functions between their classes of morphisms that preserve both units (= identities) and composition. (2) Parallel to the definition of the quasicategory CAT of all categories one can define the quasicategory CATof of all object-free categories and functors between them.

Here we list a few examples of full coreflective subcategories A of categories B. A. Modifications of the Structure 18th January 2005 Sec. 4] Subcategories 57 id X (1) Making a relation symmetric: B = Rel, A = Sym. (X, ρ ∩ ρ−1 ) −−− → (X, ρ) is an A-coreflection for (X, ρ). This example shows one of the rare instances where a subcategory is simultaneously reflective and coreflective [cf. 17(1)]. , spaces in which every sequentially closed set is closed). If (X, τ ) is a topological space, then τ = {A ⊆ X | X \ A is sequentially closed in (X, τ )} id X is a topology on X and (X, τ ) −−− → (X, τ ) is an A-coreflection for (X, τ ).

That G preserves composition follows from the uniqueness, the commutativity of the diagram F (G(B)) F (G(g)) εB   B G F (G(B )) g F (G(h)) εB GB G F (G(B ))  h εB GB and the fact that F preserves composition. Thus G is a functor. G is full because for each A-morphism f : G(B) → G(B ), the morphism εB ◦ F (f ) ◦ ε−1 B : B →B (which we denote by g) has the property that g ◦ εB = εB ◦ F (f ), and this implies [by uniqueness for (∗)] that f = G(g). G is faithful since given B g1 g2 GG B with G(g1 ) = G(g2 ) = f , an application of (∗) yields −1 −1 g1 = εB ◦ F (G(g1 )) ◦ ε−1 B = εB ◦ F (f ) ◦ εB = εB ◦ F (G(g2 )) ◦ εB = g2 .

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