## Abstract and concrete categories: the joy of catsA modern introduction to the theory of structures via the language of category theory. Unique to this book is the emphasis on concrete categories. Also noteworthy is the systematic treatment of factorization structures, which gives a new, unifying perspective to earlier work and summarizes recent developments. Each categorical notion is accompanied by many examples, usually moving from special cases to more general cases. Comprises seven chapters; the first five present the basic theory, while the last two contain more recent research results in the realm of concrete categories, cartesian closed categories and quasitopoi. The prerequisite is an elementary knowledge of set theory. Contains exercises. |

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### Contents

Introduction | 1 |

Categories Functors and Natural Transformations | 11 |

Objects and morphisms in abstract categories | 91 |

Copyright | |

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### Common terms and phrases

A-object A-reflection adjoint functor adjoint situation Ai)i Alg(T algebraic functors amnestic called cartesian closed co-adjoint cocomplete codomain coequalizer colimits commutes concrete category concrete functor concretely isomorphic conditions are equivalent considered construct coproducts coreflective coseparator defined diagram dual epi-sink epireflective essentially algebraic EXAMPLES exists a unique extremal epimorphisms extremal monomorphism factorization structure faithful fibre-small finitary finite following conditions forgetful functor free objects full concrete full subcategory function functor G G-structured Galois correspondence HComp Hence homomorphisms implies injective hull injective objects isomorphism-closed limits maps Math monadic mono-source Mono-Source)-factorization natural isomorphism natural transformation numbers pair partial morphisms poset precisely preserves Proof Proposition pullback quasicategory quotient reflective subcategory regular epimorphisms regular monomorphisms resp retraction Show Spa(T strongly complete structured arrow structured source subobjects surjective terminal object Theorem topological category topological spaces topologically algebraic unique morphism universal arrow wellpowered X-morphism