## New Model Categories From Old (1995)

Venue: | J. Pure Appl. Algebra |

Citations: | 13 - 5 self |

### BibTeX

@ARTICLE{Blanc95newmodel,

author = {David Blanc},

title = {New Model Categories From Old},

journal = {J. Pure Appl. Algebra},

year = {1995},

volume = {109},

pages = {37--60}

}

### OpenURL

### Abstract

. We review Quillen's concept of a model category as the proper setting for defining derived functors in non-abelian settings, explain how one can transport a model structure from one category to another by mean of adjoint functors (under suitable assumptions), and define such structures for categories of cosimplicial coalgebras. 1. Introduction Model categories, first introduced by Quillen in [Q1], have proved useful in a number of areas -- most notably in his treatment of rational homotopy in [Q2], and in defining homology and other derived functors in non-abelian categories (see [Q3]; also [BoF, BlS, DwHK, DwK, DwS, Goe, ScV]). From a homotopy theorist's point of view, one interesting example of such non-abelian derived functors is the E 2 -term of the mod p unstable Adams spectral sequence of Bousfield and Kan. They identify this E 2 -term as a sort of Ext in the category CA of unstable coalgebras over the mod p Steenrod algebra (see x7.4). The original purpose of this note w...

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Citation Context ...llen in [Q1], have proved useful in a number of areas -- most notably in his treatment of rational homotopy in [Q2], and in defining homology and other derived functors in non-abelian categories (see =-=[Q3]-=-; also [BoF, BlS, DwHK, DwK, DwS, Goe, ScV]). From a homotopy theorist's point of view, one interesting example of such non-abelian derived functors is the E 2 -term of the mod p unstable Adams spectr... |

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Citation Context ... which is equal to C in each dimension. When T is an additive functor between abelian categories with enough projectives, this reduces to the usual definition of derived functors (see also [Bo2, x7], =-=[DoP]-=-, [EM2], & [Hu]). We have avoided the question of when a functor will in fact preserve weak equivalences between cofibrant objects. This depends on the specific model categories in question (see Remar... |

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Citation Context ... in cohomology, and epic in cohomology through dimension n. 20 DAVID BLANC Note that since the maps p (n) are isomorphisms in degreessn, there is no lim 1 in calculating H i C = H i (lim C (n) ) (cf. =-=[Mil]-=-). This problem did not arise in the dual case (x4.2 and Proposition 5.2), since colim is exact. Note that M = R-Mod has functorial injective envelopes (constructed as in [Mac2, III, 7.4]). The constr... |

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Citation Context ...t hold for such a B (compare [BaW, 3.3, Thm. 9]). However, for the purposes of "homotopical algebra", further assumptions may be needed. In particular, in order for the "triple derived =-=functors" (cf. [BaB]-=-) of T to coincide with the right derived functors (as defined in x4.11), we would want GA to be an injective in B for any A 2 M. This will be true, for example, if all objects in M are injective (e.g... |

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Citation Context ...dgements 1.3. I would like to thank the referee for many useful comments, and in particular for suggesting Theorem 4.15 in its present generality. NEW MODEL CATEGORIES FROM OLD 3 I understand that in =-=[CaG]-=-, Cabello and Garz'on have also given conditions for defining model category structures by means of adjoint functors. 2. Model categories We begin with an exposition of Quillen's theory of model categ... |

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Citation Context ...to C in each dimension. When T is an additive functor between abelian categories with enough projectives, this reduces to the usual definition of derived functors (see also [Bo2, x7], [DoP], [EM2], & =-=[Hu]-=-). We have avoided the question of when a functor will in fact preserve weak equivalences between cofibrant objects. This depends on the specific model categories in question (see Remark 7.8 below). 8... |

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Citation Context ...nother by mean of adjoint functors (under suitable assumptions), and define such structures for categories of cosimplicial coalgebras. 1. Introduction Model categories, first introduced by Quillen in =-=[Q1]-=-, have proved useful in a number of areas -- most notably in his treatment of rational homotopy in [Q2], and in defining homology and other derived functors in non-abelian categories (see [Q3]; also [... |

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Citation Context ...motivating example of a model category is the category Top of topological spaces, with W the class of homotopy equivalences, C the class of cofibrations, and F the class of (Hurewicz) fibrations (cf. =-=[St]-=-). An alternative model category structure on Top is given in [Q1, II,x3]. However, for our purposes the basic example of a model category will be the categorysS of simplicial sets, with W the class o... |

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Citation Context ...is equal to C in each dimension. When T is an additive functor between abelian categories with enough projectives, this reduces to the usual definition of derived functors (see also [Bo2, x7], [DoP], =-=[EM2]-=-, & [Hu]). We have avoided the question of when a functor will in fact preserve weak equivalences between cofibrant objects. This depends on the specific model categories in question (see Remark 7.8 b... |

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Citation Context ...rce of a model category; in fact, it is often convenient to have additional structure, such as simplicial Hom-objects (cf. [Q1, II, x1]), properness (cf. [BoF, Def. 1.2]), and so on (see [Bau, I] and =-=[He, II] for more -=-general treatments). However, for the purposes of "homotopical algebra" -- i.e., homological algebra in non-abelian categories -- it is enough to have an RMC or an LMC (see xx2.14-2.16 below... |

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Citation Context ...ies of cosimplicial coalgebras. 1. Introduction Model categories, first introduced by Quillen in [Q1], have proved useful in a number of areas -- most notably in his treatment of rational homotopy in =-=[Q2]-=-, and in defining homology and other derived functors in non-abelian categories (see [Q3]; also [BoF, BlS, DwHK, DwK, DwS, Goe, ScV]). From a homotopy theorist's point of view, one interesting example... |

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