Abstract and concrete categories: the joy of cats by Jiri Adamek

February 23, 2017 | Construction | By admin | 0 Comments

By Jiri Adamek

This up to date introductory therapy employs class thought to discover the idea of constructions. Its new angle stresses concrete different types and provides a scientific view of factorization constructions, providing a unifying standpoint on prior paintings and summarizing contemporary advancements. quite a few examples, starting from basic to precise, remove darkness from the textual content. 1990 variation, up-to-date 2004.

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Thus h ◦ H(k) : (H ◦ F )(A) → C is an isomorphism. 18th January 2005 38 Categories, Functors, and Natural Transformations [Chap. 37 REMARK The concept of equivalence is especially useful when duality is involved. There are numerous examples of pairs of familiar categories where each category is equivalent to the dual of the other. 38 DEFINITION Categories A and B are called dually equivalent provided that Aop and B are equivalent. , to the construct of zero-dimensional compact Hausdorff spaces and continuous maps).

An equivalence can be obtained by associating with each set its power-set, considered as a complete atomic boolean algebra. (4) The category of compact Hausdorff abelian groups is dually equivalent to Ab. An equivalence can be obtained by associating with each compact Hausdorff abelian group G its group of characters hom(G, / ) (Pontrjagin Duality). ❘❩ (5) The category of locally compact abelian groups is dually equivalent to itself. An equivalence can be obtained as in (4) above. (6) The category HComp of compact Hausdorff spaces (and continuous functions) is dually equivalent to the category of C ∗ -algebras and algebra homomorphisms.

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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