## Categories for the working mathematician |

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### Contents

Categories Functors and Natural Transformations | 7 |

Constructions on Categories | 31 |

Universals and Limits | 55 |

Copyright | |

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### Common terms and phrases

abelian category abelian groups adjoint functor adjoint functor theorem adjunction F algebra arrow h assigns axioms bifunctor bijection binary biproduct called CGHaus codomain coend cokernel colimits comma category commutative diagram components composite construction continuous maps coproduct counit defined definition described dinatural domain dual elements equalizer equivalence example Exercises exists factors forgetful functor full subcategory functor category functor F given graph Hausdorff spaces hence hom-sets homomorphism identity arrow implies initial object injective inverse kernel left adjoint limiting cone Mac Lane monad monic monoidal category morphism natural isomorphism natural transformation object function pair of arrows parallel pair preorder projections Proof Proposition prove pullback quotient R-Mod R-module right adjoint right Kan extension ring simplicial small hom-sets small set small-complete square commutes subobjects subset T-algebras tensor product terminal object topological space unique arrow unit universal arrow usual vector space vertex