## Stabilization of model categories (1998)

Citations: | 222 - 9 self |

### BibTeX

@MISC{Hovey98stabilizationof,

author = {Mark Hovey},

title = {Stabilization of model categories},

year = {1998}

}

### Years of Citing Articles

### OpenURL

### Abstract

monoidal structure which is compatible with the model structure. Given a monoidal model category, we consider the homotopy theory of modules over a given monoid and the homotopy theory of monoids. We make minimal assumptions on our model categories; our results therefore are more general, yet weaker, than the results of [10]. In particular, our results apply to the monoidal model category of topological symmetric spectra [7].

### Citations

368 | Homotopical algebra - Quillen - 1967 |

160 |
modules, and algebras in stable homotopy theory
- Elmendorf, Kriz, et al.
- 1997
(Show Context)
Citation Context ...enerated topological spaces, and chain complexes of modules over a commutative ring. The thirty-year long search for a monoidal model category of spectra met success with the category of S-modules of =-=[3]-=- and the symmetric spectra of [7]. Given any monoidal category, one has categories of monoids and of modules over a given monoid. If we are working in a monoidal model category, we would like these as... |

152 |
Model categories, Mathematical Surveys and Monographs 63
- Hovey
- 1999
(Show Context)
Citation Context ...l categories of modules over a monoid A in a cofibrantly generated monoidal model category C. For cofibrantly generated model categories, and for all terms left undefined here, we refer the reader to =-=[5]-=-; cofibrantly generated model categories appear in Section 2.1. Monoidal model categories are discussed in [5, Chapter 4], but we will remind the reader of the basic definition. In a monoidal category... |

149 | Algebras and modules in monoidal model categories
- Schwede, Shipley
- 2000
(Show Context)
Citation Context ... theory of modules over a given monoid and the homotopy theory of monoids. We make minimal assumptions on our model categories; our results therefore are more general, yet weaker, than the results of =-=[10]-=-. In particular, our results apply to the monoidal model category of topological symmetric spectra [7]. Introduction A monoidal model category is a (closed) monoidal category that is also a model cate... |

135 |
Homotopy theories and model categories, in Handbook of Algebraic Topology
- Dwyer, Spalinski
- 1995
(Show Context)
Citation Context ... object in question is cofibrant in A-mod. Nevertheless, we can follow the usual definitions in the standard construction of the homotopy category of a model category. See for example [5, Chapter 1], =-=[2]-=-, or [4, Chapters 8 and 9]. In order for the notion of left homotopy to have any content, one must assume that the sources of one’s maps are cofibrant in A-mod. Similarly, for right homotopy, one must... |

38 | The stable category and generalized Thom spectra - Lewis - 1978 |

12 |
Model categories and general abstract homotopy theory
- Dwyer, Hirschhorn, et al.
(Show Context)
Citation Context ... categories on the part of the reader. A gentle introduction to the subject can be found in [DS95]. A more thorough and highly recommended source is [Hir97, Part 2]. Other sources include [Hov97] and =-=[DHK]-=-. In particular, in a model category C, we have a cofibrant replacement functor Q and a fibrant replacement functor R. There is a natural trivial fibration QX q −→ X, and QX is cofibrant. Similarly, t... |

6 |
Symmetric spectra, preprint
- Hovey, Shipley, et al.
- 1998
(Show Context)
Citation Context ...on our model categories; our results therefore are more general, yet weaker, than the results of [10]. In particular, our results apply to the monoidal model category of topological symmetric spectra =-=[7]-=-. Introduction A monoidal model category is a (closed) monoidal category that is also a model category in a compatible way. Monoidal model categories abound in nature: examples include simplicial sets... |

4 |
Localization, cellularization, and homotopy colimits, preprint
- Hirschhorn
(Show Context)
Citation Context ...thermore, f □g is a weak equivalence if either f or g is. However, we will have trouble with the unit unless we assume either A or S is cofibrant, just as above. Since Hirschhorn’s landmark treatment =-=[4]-=-, it has become clear that the right collection of model categories to work with is the collection of left proper cellular model categories. Hirschhorn shows that one can perform Bousfield localizatio... |

2 |
Model categories, preprint (x+194
- Hovey
- 1997
(Show Context)
Citation Context ...y with model categories on the part of the reader. A gentle introduction to the subject can be found in [DS95]. A more thorough and highly recommended source is [Hir97, Part 2]. Other sources include =-=[Hov97]-=- and [DHK]. In particular, in a model category C, we have a cofibrant replacement functor Q and a fibrant replacement functor R. There is a natural trivial fibration QX q −→ X, and QX is cofibrant. Si... |