## Simplicial Structures on Model Categories and Functors (2001)

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Venue: | Amer.J.Math.123 |

Citations: | 16 - 3 self |

### BibTeX

@ARTICLE{Rezk01simplicialstructures,

author = {Charles Rezk and Stefan Schwede and Brooke Shipley},

title = {Simplicial Structures on Model Categories and Functors},

journal = {Amer.J.Math.123},

year = {2001},

volume = {123},

pages = {551--575}

}

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### Abstract

We produce a highly structured way of associating a simplicial category to a model category which improves on work of Dwyer and Kan and answers a question of Hovey. We show that model categories satisfying a certain axiom are Quillen equivalent to simplicial model categories. A simplicial model category provides higher order structure such as composable mapping spaces and homotopy colimits. We also show that certain homotopy invariant functors can be replaced by weakly equivalent simplicial, or "continuous," functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence.

### Citations

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Citation Context ...unique up to simplicial Quillen equivalence. 1. Introduction. In [DK] Dwyer and Kan showed that a simplicial category, called the hammock localization, can be associated to any Quillen model category =-=[Qui]-=-. This simplicial category captures higher order information, for example fibration and cofibration sequences and mapping spaces, see [Qui, I 3], which is not captured by the ordinary homotopy categor... |

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Citation Context ... j ; j is a simplicial functor we show that K C jXj is isomorphic to jK sC Xj. Here C and sC are the simplicial actions in the respective categories. These are not to be confused with the coends, see =-=[ML], ∆ -=-and ∆ ∆ which follow. Since the left adjoint j ; j is a strong simplicial functor, that is, the natural transformation is an isomorphism, it follows that the right adjoint Sing is also a simplicia... |

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(Show Context)
Citation Context ... a level equivalence since C is stable. Differential graded modules. A cofibrantly generated model category, D, of differential graded modules over a differential graded algebra, A, is constructed in =-=[SSa, 5]-=-, see also [Hov, 2.3.11]. The weak equivalences and fibrations are the quasi-isomorphisms and surjections of the underlying Z-graded chain complexes. Since D is stable and proper, the realization axio... |

58 |
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(Show Context)
Citation Context ...l, or “continuous,” functors. This is used to show that if a simplicial model category structure exists on a model category then it is unique up to simplicial Quillen equivalence. 1. Introduction.=-= In [DK]-=- Dwyer and Kan showed that a simplicial category, called the hammock localization, can be associated to any Quillen model category [Qui]. This simplicial category captures higher order information, fo... |

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Citation Context ... which is proved as Corollary 6.2. Assume that C and D are model categories which either satisfy the hypotheses of Theorem 1.1 or satisfy the hypotheses of one of the general localization machines in =-=[Hir]-=- or [Smi], see also [Dug]. Manuscript received February 7, 2000; revised August 3, 2000. Research of the first author supported in part by an AMS Centennial Fellowship; research of the third author su... |

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Citation Context ...lary 6.2. Assume that C and D are model categories which either satisfy the hypotheses of Theorem 1.1 or satisfy the hypotheses of one of the general localization machines in [Hir] or [Smi], see also =-=[Dug]-=-. Manuscript received February 7, 2000; revised August 3, 2000. Research of the first author supported in part by an AMS Centennial Fellowship; research of the third author supported in part by NSF gr... |

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(Show Context)
Citation Context ...d answers another question of Hovey [Hov, 8.9]. For a model category C, our candidate for a Quillen equivalent simplicial model category is based on the category of simplicial objects in C, sC. Reedy =-=[Ree]-=- establishes the Reedy model category on sC, but it is neither simplicial nor Quillen equivalent to C, see [DKS, 2.6] or Corollary 7.4. So we localize the Reedy model category to create the realizatio... |

11 |
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(Show Context)
Citation Context ... the axioms for the realization model category on sC we assume that C is a cofibrantly generated model category. We now recall a version of the definition of cofibrantly generated model category from =-=[DHK]-=-, or see [Hov, 2.1.17], [SSa, 2.2], or [Hir]. For a cocomplete category C and a class I of maps, the I-injectives are the maps with the right lifting property with respect to the maps in I. TheI-cofib... |

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Citation Context ...ial sets. LEMMA 4.3. The model category of simplicial sets, S, satisfies Realization Axiom 3.4. Below we verify that Lemma 4.3 is a special case of the following proposition, essentially due to Puppe =-=[Pup]-=-. PROPOSITION 4.4. Let I be a small category and X ! Y be a map of I-diagrams of simplicial sets such that for each i1 ! i2 2 I the square X(i1) ;;;! Y(i1) ? ? y ? y X(i2) ;;;! Y(i2)s560 CHARLES REZK,... |

3 |
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(Show Context)
Citation Context ... proved as Corollary 6.2. Assume that C and D are model categories which either satisfy the hypotheses of Theorem 1.1 or satisfy the hypotheses of one of the general localization machines in [Hir] or =-=[Smi]-=-, see also [Dug]. Manuscript received February 7, 2000; revised August 3, 2000. Research of the first author supported in part by an AMS Centennial Fellowship; research of the third author supported i... |

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1 |
Rezk, Fibrations and homotopy colimits of simplicial sheaves
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Citation Context ...alence by Remark 3.2. Therefore, for such f and for every i 2 ∆ the map X(i) ! Y(i) isa weak equivalence, i.e., f is a level weak equivalence. A proof of Proposition 4.4 in this generality appears i=-=n [Rez]-=- where it is generalized to simplicial sheaves. Alternatively, one can adapt the argument of [Far, App. HL], where the Proposition is stated under the additional hypothesis that the nerve of the index... |