Basic Category Theory
At the heart of this short introduction to category theory is the idea of a universal property, important throughout mathematics. After an introductory chapter giving the basic definitions, separate chapters explain three ways of expressing universal properties: via adjoint functors, representable functors, and limits. A final chapter ties all three together. The book is suitable for use in courses or for independent study. Assuming relatively little mathematical background, it is ideal for beginning graduate students or advanced undergraduates learning category theory for the first time. For each new categorical concept, a generous supply of examples is provided, taken from different parts of mathematics. At points where the leap in abstraction is particularly great (such as the Yoneda lemma), the reader will find careful and extensive explanations. Copious exercises are included.
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Categories functors and natural transformations
Interlude on sets
Adjoints representables and limits
Appendix Proof of the general adjoint functor theorem
adjoint functor adjoint functor theorem adjunction Aop,Set axioms bijection binary products called canonical cartesian closed category of sets category theory comma category commutes composition construct continuous maps Corollary deﬁne defined definition diagram dual epic equal equations equivalence relation exactly Example Exercise exists a unique forgetful functor full subcategory functor categories functor F functors A F GF(A given homomorphism inclusion initial object inverse isomorphism classes left adjoint lim←I D limit cone limits and colimits limits of shape linear map locally small category map f mathematics monic monoid natural isomorphism natural numbers natural transformation notation one-object category ordered set pair poset preserves limits presheaf Proof Proposition prove pullback right adjoint ring satisfying set theory sets and functions Similarly subobject subsets surjective terminal object topological space triangle identities unique map universal property Vectk vector space weakly initial set write Yoneda lemma