## Category TheoryCategory theory is a branch of abstract algebra with incredibly diverse applications. This text and reference book is aimed not only at mathematicians, but also researchers and students of computer science, logic, linguistics, cognitive science, philosophy, and any of the other fields in which the ideas are being applied. Containing clear definitions of the essential concepts, illuminated with numerous accessible examples, and providing full proofs of all important propositions and theorems, this book aims to make the basic ideas, theorems, and methods of category theory understandable to this broad readership. Although assuming few mathematical pre-requisites, the standard of mathematical rigour is not compromised. The material covered includes the standard core of categories; functors; natural transformations; equivalence; limits and colimits; functor categories; representables; Yoneda's lemma; adjoints; monads. An extra topic of cartesian closed categories and the lambda-calculus is also provided - a must for computer scientists, logicians and linguists! This Second Edition contains numerous revisions to the original text, including expanding the exposition, revising and elaborating the proofs, providing additional diagrams, correcting typographical errors and, finally, adding an entirely new section on monoidal categories. Nearly a hundred new exercises have also been added, many with solutions, to make the book more useful as a course text and for self-study. |

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### Contents

1 Categories | 1 |

2 Abstract structures | 29 |

3 Duality | 53 |

4 Groups and categories | 75 |

5 Limits and colimits | 89 |

6 Exponentials | 119 |

7 Naturality | 147 |

8 Categories of diagrams | 185 |

9 Adjoints | 207 |

10 Monads and algebras | 253 |

Solutions to selected exercises | 279 |

303 | |

305 | |

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### Common terms and phrases

A X B adjunction arrow f bijective binary products Boolean algebra cartesian closed category category theory cocomplete coequalizer colimit composition consider constructed contravariant coproduct counit defined definition determined diagram commutes dual duality Eacample elements endofunctor equations equivalence relation example exponential following diagram forgetful functor free monoid function f functor category functor F given graph Heyting algebra Hom(A Hom(C Hom(X HomC homomorphism h I-indexed identity arrows implies initial object injective inverse left adjoint limits locally small logic monad monic mono monoidal category morphism natural isomorphism natural numbers natural transformation notion objects and arrows operations pair poset powerset preserves projection Proof Proposition pullback quotient representable functor right adjoint SetsCop SetsI Similarly slice category small category structure subset suppose surjective T-algebra terminal object theorem ultrafilter unique Yoneda embedding Yoneda lemma