## Categorical Homotopy TheoryThis book develops abstract homotopy theory from the categorical perspective with a particular focus on examples. Part I discusses two competing perspectives by which one typically first encounters homotopy (co)limits: either as derived functors definable when the appropriate diagram categories admit a compatible model structure, or through particular formulae that give the right notion in certain examples. Riehl unifies these seemingly rival perspectives and demonstrates that model structures on diagram categories are irrelevant. Homotopy (co)limits are explained to be a special case of weighted (co)limits, a foundational topic in enriched category theory. In Part II, Riehl further examines this topic, separating categorical arguments from homotopical ones. Part III treats the most ubiquitous axiomatic framework for homotopy theory - Quillen's model categories. Here, Riehl simplifies familiar model categorical lemmas and definitions by focusing on weak factorization systems. Part IV introduces quasi-categories and homotopy coherence. |

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### Contents

All concepts are Kan extensions | 3 |

Derived functors via deformations | 17 |

Basic concepts of enriched category theory | 32 |

The unreasonably effective cobar construction | 58 |

The theory | 69 |

The practice | 76 |

Weighted limits and colimits | 99 |

Categorical tools for homotopy colimit computations | 121 |

Algebraic perspectives on the small object argument | 190 |

Enriched factorizations and enriched lifting properties | 222 |

A brief tour of Reedy category theory | 240 |

Preliminaries on quasicategories | 263 |

Simplicial categories and homotopy coherence | 282 |

Isomorphisms in quasicategories | 298 |

A sampling of 2categorical aspects of quasicategory theory | 318 |

337 | |

Weighted homotopy limits and colimits | 136 |

Derived enrichment | 145 |

FACTORIZATION SYSTEMS | 167 |

Glossary of Notation | 343 |

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### Common terms and phrases

1-simplices algebraic weak factorization arrows bar construction bifunctor canonical categorical equivalence category theory closed symmetric monoidal cocomplete cofibrant objects cofibrant replacement cofibrantly commutative comonad composite coproduct Corollary cosimplicial defined definition diagram enriched category Example fibrant objects fibrations functor tensor product functorial factorization geometric realization Hence hom-objects hom-sets hom-spaces homotopy category homotopy coherent homotopy colimit internal hom Kan extension left adjoint left deformation left derived functor left Kan extension lifting problem lifting property limits and colimits model structure monad monomorphism morphisms n-simplex natural transformation non-degenerate pointwise cofibrant preserves proof pullback pushout quasi-categories Reedy cofibrant Remark right adjoint simplicial category simplicial model category simplicial object simplicial set simplicially enriched small object argument sSet sSet+ subcategory terminal object Theorem topological trivial cofibrations trivial fibration two-variable adjunction underlying category unenriched universal property V-category V-functor vertices weak equivalences weak factorization system weak homotopy equivalences weighted limits Yoneda lemma