## Codescent theory II: Cofibrant approximations (2003)

Citations: | 2 - 2 self |

### BibTeX

@TECHREPORT{Balmer03codescenttheory,

author = {Paul Balmer and Michel Matthey},

title = {Codescent theory II: Cofibrant approximations},

institution = {},

year = {2003}

}

### OpenURL

### Abstract

Abstract. We establish a general method to produce cofibrant approximations in the model category US(C, D) of S-valued C-indexed diagrams with D-weak equivalences and D-fibrations. We also present explicit examples of such approximations. Here, S is an arbitrary cofibrantly generated simplicial model category and D ⊂ C are small categories. An application to the notion of homotopy colimit is presented. 1.

### Citations

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(Show Context)
Citation Context ...the sequel and presents some basic facts concerning categories of diagrams. It also contains a little summary of what is needed from Part I on the model category US(C, D). We refer to Mac Lane’s book =-=[14]-=- for purely categorical questions. We refer to Hirschhorn [10] or to Hovey [11] for model category questions. Appendix A of Part I gives a concise list of prerequisites. The notion of simplicial model... |

381 |
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(Show Context)
Citation Context ...e to a question regarding homotopy colimits, that many topologists might have asked themselves once. Two approaches to homotopy colimits are available, both originating from the work of Bousfield-Kan =-=[4]-=-. First, one can think of the homotopy colimit basically as an esoteric but explicit formula which one can apply to whatever moves around, say, in any category with an action of simplicial sets. The s... |

375 |
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(Show Context)
Citation Context ...otopy Lifting Extension Theorem for categories of simplicial objects (in some category) enriched over sSets, see [12]. This was then taken as Axiom (SM7) for a simplicial model category by Quillen in =-=[15]-=- (see Axiom (4) in Definition A.1). Definition 4.1. Let S1, S2 and S3 be model categories. Let ⊙: S1 × S2 −→ S3 be a right tensorial coupling, with right mapping functor map ⊙. We say that ⊙ has the c... |

375 |
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(Show Context)
Citation Context ...category C, we denote by BC = B•C ∈ sSets its nerve, whose realization |BC| ∈ Top is the usual classifying space of C. Here, we follow Segal’s modern definition of the nerve in [17], see also Quillen =-=[16]-=- (and not Bousfield-Kan’s old definition in [4], where their BC is our B(C op ); note however that |BC| and |B(C op )| are canonically homeomorphic). Notation 6.1. Let D be a subcategory of a small ca... |

227 | Model categories
- Hovey
- 1999
(Show Context)
Citation Context ...also contains a little summary of what is needed from Part I on the model category US(C, D). We refer to Mac Lane’s book [14] for purely categorical questions. We refer to Hirschhorn [10] or to Hovey =-=[11]-=- for model category questions. Appendix A of Part I gives a concise list of prerequisites. The notion of simplicial model category being central here, it is recalled in Appendix A of the present Part.... |

204 |
Model categories and their localizations
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- 2003
(Show Context)
Citation Context ..., which we have denoted US(C, D), where the weak-equivalences and the fibrations are defined D-objectwise only, on a subset of objects D ⊂ C, see [1, Thm 3.5]. The reader of Dugger [6] and Hirschhorn =-=[10]-=- can as well keep the absolute case D = C in mind. Unfortunately, these left model constructions, although very popular, usually leave the cofibrations mysterious. The factorization axiom in S C , whi... |

192 | Simplicial Homotopy Theory - Goerss, Jardine - 1999 |

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(Show Context)
Citation Context ...rt with, given a small category C, we denote by BC = B•C ∈ sSets its nerve, whose realization |BC| ∈ Top is the usual classifying space of C. Here, we follow Segal’s modern definition of the nerve in =-=[17]-=-, see also Quillen [16] (and not Bousfield-Kan’s old definition in [4], where their BC is our B(C op ); note however that |BC| and |B(C op )| are canonically homeomorphic). Notation 6.1. Let D be a su... |

39 |
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(Show Context)
Citation Context ... – and to use – that whatever holds for diagrams in sSets remains true for diagrams in familiar model categories, now have a rigorous argument at their disposal. As already observed in Hollender-Vogt =-=[13]-=- in the special case of topological spaces, the above diagram E is of central importance for homotopy colimits. The last short section of the paper is an application of the above to a question regardi... |

38 | Universal homotopy theories
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(Show Context)
Citation Context ...l structure on S C , which we have denoted US(C, D), where the weak-equivalences and the fibrations are defined D-objectwise only, on a subset of objects D ⊂ C, see [1, Thm 3.5]. The reader of Dugger =-=[6]-=- and Hirschhorn [10] can as well keep the absolute case D = C in mind. Unfortunately, these left model constructions, although very popular, usually leave the cofibrations mysterious. The factorizatio... |

34 | Algebraic K-theory and étale - Thomason - 1985 |

22 |
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- 2002
(Show Context)
Citation Context ...trary model category S, there is in general no known model structure on C-indexed diagrams S C , with objectwise weak equivalences. Hence the notion of “model approximation” of Chachólski and Scherer =-=[5]-=-, that is not used here. Nevertheless, it is well-known that if the model category S is cofibrantly generated, one can create a so-called left model structure on S C , denoted US(C) hereafter, by defi... |

14 |
Function complexes for diagrams of simplicial sets
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- 1983
(Show Context)
Citation Context ...is given by Lemma 5.2; then, ϑ and E fulfill the required properties needed in Theorem 5.4. The point here is that the model category UsSets(D op × D) is à la Dwyer-Kan-Heller-Dugger as considered in =-=[7, 9, 6]-=-, that is, with sSets as category of “values” and without need of a relative model category structure in the sense of 2.6. See also Remark 6.12 below. Remark 5.6. The only property of the morphism ηX ... |

6 | Model theoretic reformulation of the BaumConnes and Farrell-Jones conjectures
- Balmer, Matthey
(Show Context)
Citation Context ... with values in some model category S, like for instance the category of (compactly generated Hausdorff) topological spaces. One recent illustration of this importance, among many others, is given in =-=[2]-=-, where we show that the K-theoretic Isomorphism Conjectures boil down to understanding cofibrant approximations in a suitable category of diagrams. In this spirit, cofibrant approximations might be t... |

2 | Codescent theory I: Foundations - Balmer, Matthey - 2003 |

1 |
On c.s.s. categories
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- 1957
(Show Context)
Citation Context ... the next definition turns crucial. The origin of such a concept goes back to Kan’s Homotopy Lifting Extension Theorem for categories of simplicial objects (in some category) enriched over sSets, see =-=[12]-=-. This was then taken as Axiom (SM7) for a simplicial model category by Quillen in [15] (see Axiom (4) in Definition A.1). Definition 4.1. Let S1, S2 and S3 be model categories. Let ⊙: S1 × S2 −→ S3 b... |