Categories for the Working MathematicianCategory Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunctors with a natural isomorphism between corresponding sets of arrows. All these forms, with their interrelations, are examined in Chapters III to V. The slogan is "Adjoint functors arise everywhere". Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general ized monoid. Chapters VI and VII explore this notion and its generaliza tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure (given by a tensor product) lead inter alia to the study of more convenient categories of topological spaces. |
Contents
Axioms for Categories | 1 |
Categories | 12 |
Monics Epis and Zeros | 19 |
Copyright | |
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A(Fx Ab-category abelian category abelian groups absolute coequalizer adjoint functor adjoint functor theorem adjunction F algebra arrow f assigns axioms bifunctor bijection binary canonical cartesian closed CGHaus codomain coend Colim colimits comma category comparison functor composite construction contravariant coproduct counit defined definition dinatural dual elements equalizer equivalence example Exercises exists factors forgetful functor full subcategory functor category functor F given graph Hausdorff spaces hence hom-sets identity arrow implies initial object inverse Kan extension kernel left adjoint Lemma Lim F limiting cone monad monic monoidal category morphism natural isomorphism natural transformation pair of arrows parallel pair preorder projections Proof Proposition prove pullback quotient R-Mod R-module right adjoint right Kan extension ring small hom-sets small set small-complete subobjects subset T-algebras tensor product topological space topology unique arrow universal arrow usual vector space vertex