## Category TheoryThis text and reference book on Category Theory, a branch of abstract algebra, is aimed not only at students of Mathematics, but also researchers and students of Computer Science, Logic, Linguistics, Cognitive Science, Philosophy, and any of the other fields that now make use of it. 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 it assumes 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. |

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### Contents

2 | 23 |

3 | 45 |

4 | 63 |

Exponentials | 105 |

Functors and naturality | 125 |

Categories of diagrams | 159 |

Adjoints | 179 |

Monads and algebras | 223 |

249 | |

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### Common terms and phrases

adjoint functors adjunction arrow f axioms bijective binary products Boolean algebra called cartesian closed category category theory cocomplete coequalizer colimits comonad composition condition congruence consider construction contravariant coproduct counit deﬁned deﬁnition diﬀerent dual duality elements endofunctor equalizer equations equivalence classes equivalence relation example exercise exponential finite ﬁrst forgetful functor free monoid function f functor category functor F given graph Heyting algebra homomorphism h identity arrow iﬀ implies inﬁnite initial algebra initial object injective inverse left adjoint limits locally small logic mathematical monad monic monotone morphism natural isomorphism natural numbers natural transformation notion objects and arrows operation pair poset powerset preserves Proof pullback reader representable functor right adjoint sense Sets Cop Sets/I Similarly slice category small category Speciﬁcally structure subsets surjective T-algebra terminal object theorem topological space ultraﬁlter unique Yoneda embedding Yoneda Lemma