## An Introduction to Category TheoryCategory theory provides a general conceptual framework that has proved fruitful in subjects as diverse as geometry, topology, theoretical computer science and foundational mathematics. Here is a friendly, easy-to-read textbook that explains the fundamentals at a level suitable for newcomers to the subject. Beginning postgraduate mathematicians will find this book an excellent introduction to all of the basics of category theory. It gives the basic definitions; goes through the various associated gadgetry, such as functors, natural transformations, limits and colimits; and then explains adjunctions. The material is slowly developed using many examples and illustrations to illuminate the concepts explained. Over 200 exercises, with solutions available online, help the reader to access the subject and make the book ideal for self-study. It can also be used as a recommended text for a taught introductory course. |

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### Contents

1 | |

2 Basic gadgetry | 34 |

3 Functors and natural transformations | 72 |

4 Limits and colimits in general | 108 |

5 Adjunctions | 148 |

6 Posets and monoid sets | 190 |

223 | |

224 | |

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### Common terms and phrases

abelian group adjunction algebraic arbitrary category arrow f bijection binary products cartesian product category theory coequalizer cofree complete posets component composition continuous map contravariant functor coproduct counit covariant functor deﬁned Deﬁnition Deﬁnition Let diagram diagram chase edge element endo-functor epic examples Exercises f and g ﬁnal object ﬁnd ﬁnite ﬁrst forgetful functor function f furnished gadgets given gives hom-functor identity arrows inverse pair isomorphism left adjoint left hand left solution Lemma lower sections monic monoid morphism monotone function monotone map natural transformations nodes notation notion object assignment pair of arrows parallel pair poset preset presheaves produce proof pullback R-morphism R-set right adjoint right solution section we look Set-R Show square commutes structured sets subset template Theorem topological spaces triangle commutes trivial unique arrow universal solution V-diagram wedge