## Combinatorial model categories have presentations (2001)

Venue: | Adv. in Math. 164 |

Citations: | 50 - 7 self |

### BibTeX

@INPROCEEDINGS{Dugger01combinatorialmodel,

author = {Daniel Dugger},

title = {Combinatorial model categories have presentations},

booktitle = {Adv. in Math. 164},

year = {2001}

}

### OpenURL

### Abstract

Abstract. We show that every combinatorial model category is Quillen equivalent to a localization of a diagram category (where ‘diagram category’ means diagrams of simplicial sets). This says that every combinatorial model

### Citations

327 | Homotopical algebra - Quillen - 1967 |

307 |
Homotopy limits, completions and localizations
- Bousfield, Kan
- 1972
(Show Context)
Citation Context ...ven a simplicial diagram F: ∆ op → UC. Since UC is a simplicial model category we may form the geometric realization |F |. On the other hand we may also form the homotopy colimit using the formula in =-=[BK]-=-, and we will denote this object by ‘badhocolimF’. The ‘bad-’ prefix is to remind us that this is not a priori a homotopy invariant construction, because the objects in the diagram F need not be cofib... |

204 | Model Categories - Hovey - 1999 |

197 |
Locally Presentable and Accessible Categories
- Adámek, Rosick´y
- 1994
(Show Context)
Citation Context ...s of A. For background on locally presentable categories one may consult [AR, Section 1.B] or [B]. The condition that an object be small with respect to λ-filtered colimits is called λ-presentable in =-=[AR]-=-, but we will follow Smith and call it λ-small. Locally presentable categories have the following important characteristics: (1) For every object A, there exists a regular cardinal λ such that A is λ-... |

64 | Accessible categories: the foundations of categorical model theory, volume 104 of Contemporary Mathematics - Makkai, Paré - 1989 |

37 | Universal homotopy theories
- Dugger
(Show Context)
Citation Context ...ry’ means diagrams of simplicial sets). This says that every combinatorial model category can be built from a category of ‘generators’ and a set of ‘relations’. 1. Introduction In the companion paper =-=[D2]-=- we introduced a technique for constructing model categories via generators and relations. The two main points were as follows: (1) From a small category C one can construct a model category UC which ... |

30 |
Model categories and more general abstract homotopy theory, Book in preparation, available at http://www-math.mit.edu/ ∼ psh
- Dwyer, Hirschhorn, et al.
(Show Context)
Citation Context ...and A K (in fact all this needs is that M is complete and co-complete). The tensoring operation is used in the proof of Lemma 6.4, and it’s also the basis for the way homotopy colimits are defined in =-=[DHK]-=- and [H]. We briefly recall this definition of homotopy colimits, and we list some basic properties which aren’t always stressed. These properties are used to prove Proposition A.4, which is a techniq... |

28 |
Localization of model categories
- Hirschhorn
- 1999
(Show Context)
Citation Context ...n fact all this needs is that M is complete and co-complete). The tensoring operation is used in the proof of Lemma 6.4, and it’s also the basis for the way homotopy colimits are defined in [DHK] and =-=[H]-=-. We briefly recall this definition of homotopy colimits, and we list some basic properties which aren’t always stressed. These properties are used to prove Proposition A.4, which is a technique for i... |

21 |
Homotopy theories
- Heller
- 1988
(Show Context)
Citation Context ... UC/S. We must work a little harder to show that M is Quillen equivalent to a model category in which every object is cofibrant. Recall that the diagram category sSet Cop has a Heller model structure =-=[He]-=- in which a map D → E is a weak equivalence (resp. cofibration) if D(c) → E(c) is a weak equivalence (resp. cofibration) for every c ∈ C. The Heller model structure is related to UC by a Quillen equiv... |

13 |
Handbook of Categorical Algebra. II. Categories and Structures
- Borceux
- 1994
(Show Context)
Citation Context ...ect to λ-filtered colimits, and (ii) Every object of M can be expressed as a λ-filtered colimit of elements of A. For background on locally presentable categories one may consult [AR, Section 1.B] or =-=[B]-=-. The condition that an object be small with respect to λ-filtered colimits is called λ-presentable in [AR], but we will follow Smith and call it λ-small. Locally presentable categories have the follo... |

4 |
Replacing model categories by simplicial ones
- Dugger
(Show Context)
Citation Context ...lowing corollary: Corollary 1.2. Every combinatorial model category is Quillen equivalent to one which is simplicial, left proper, and (this is slightly harder) in which every object is cofibrant. In =-=[D1]-=- it was proven, using very different methods, that every left proper, combinatorial model category is Quillen equivalent to a simplicial model category. The above corollary offers a slight improvement... |

1 |
Combinatorial model categories. To exist
- Smith
(Show Context)
Citation Context ...uch set. The following proposition brings together the properties of combinatorial model categories we will need in this paper. Most of these statements are due to Smith, and should one day appear in =-=[Sm]-=-. For the reader’s convenience we provide proofs (or sketches of proofs, when we are lazy) in section 7. Proposition 2.3. Let M be a combinatorial model category. (i) There exist cofibrant- and fibran... |