## An Introduction to the Language of Category TheoryThis textbook provides an introduction to elementary category theory, with the aim of making what can be a confusing and sometimes overwhelming subject more accessible. In writing about this challenging subject, the author has brought to bear all of the experience he has gained in authoring over 30 books in university-level mathematics. The goal of this book is to present the five major ideas of category theory: categories, functors, natural transformations, universality, and adjoints in as friendly and relaxed a manner as possible while at the same time not sacrificing rigor. These topics are developed in a straightforward, step-by-step manner and are accompanied by numerous examples and exercises, most of which are drawn from abstract algebra. The first chapter of the book introduces the definitions of category and functor and discusses diagrams,duality, initial and terminal objects, special types of morphisms, and some special types of categories,particularly comma categories and hom-set categories. Chapter 2 is devoted to functors and naturaltransformations, concluding with Yoneda's lemma. Chapter 3 presents the concept of universality and Chapter 4 continues this discussion by exploring cones, limits, and the most common categorical constructions – products, equalizers, pullbacks and exponentials (along with their dual constructions). The chapter concludes with a theorem on the existence of limits. Finally, Chapter 5 covers adjoints and adjunctions. Graduate and advanced undergraduates students in mathematics, computer science, physics, or related fields who need to know or use category theory in their work will find An Introduction to Category Theory to be a concise and accessible resource. It will be particularly useful for those looking for a more elementary treatment of the topic before tackling more advanced texts. |

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I've found two errors in equations and use of symbols within the first few pages. Although it's written well it's annoying. Time will tell if my constant unease at whether I'm reading something incorrect will lead to me taking more care to understand - if only to check the author's work.

However, I bought this as an easy light reference and it's not going to serve that purpose because with the mistakes you can't really use it as a reference.

### Contents

1 | |

Functors and Natural Transformations | 37 |

Universality | 71 |

Cones and Limits | 87 |

Adjoints | 119 |

Answers to Selected Exercises
| 145 |

163 | |

165 | |

166 | |

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¼ f AbGrp adjoint functor arrow bijections called category theory cocone codomain coequalizer comma category commutes composition concepts contravariant functor coproduct couniversal covariant functor ð Þ defined Definition Let denoted diagram dual elements epic equalizer Example f ¼ f and g forgetful functor free group function f functor category functor F fusion formulas group homomorphism hom functor hom-set homC homc(A home(C hompſ identity functor identity morphism implies inclusion map initial object injective inverse labeled left adjoint Let F mediating morphism map monic morphism f natural isomorphism natural transformation nodes nonempty numbers Poset projection maps pullback quasiuniversal right-invertible ring sends set function Show shown in Figure surjective terminal cone terminal object Theorem Tºp unique mediating morphism universal pair Vect vector spaces zero object Þ ¼ Þ¼