Quillen model structures for relative homological algebra

Christensen, J. Daniel; Hovey, Mark

doi:10.1017/s0305004102006126

Cited by 71 publications

(74 citation statements)

References 14 publications

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“…In this last section we give a characterisation of the weak fibrations and weak cofibrations of unbounded chain complexes in an abelian category A that one gets when applying our definition to the model structures defined in [CH02]. We start by fixing some notations and giving a short description of these model structures.…”

Section: The Case Of Chain Complexes In An Abelian Categorymentioning

confidence: 99%

“…In the article [CH02], model structures are defined on Ch · (A) with respect to a given projective class; this consists of a class of A-objects one thinks of as the class of projective objects, together with a class of A-maps one thinks of as the class of epimorphisms. This notion was originally introduced by Maranda in [Mar64].…”

Section: The Case Of Chain Complexes In An Abelian Categorymentioning

confidence: 99%

“…Theorem 2.2 of [CH02] gives hypotheses for the classes of P-fibrations, Pcofibrations and P-equivalences to form a model structure…”

Section: The Case Of Chain Complexes In An Abelian Categorymentioning

confidence: 99%

“…Clearly, all of its objects are fibrant, and any chain homotopy equivalence is a P-equivalence. Proposition 2.5 of [CH02] states that a map is a P-cofibration exactly when it is a degreewise split monomorphism with a Pcofibrant cokernel, and Lemma 2.4 that a chain complex C · is P-cofibrant if and only if each C n is P-projective and every map from C · to a weakly P-contractible object K · is nullhomotopic. A detailed and direct proof that Ch · (R) with the categorical projective class on R Mod-the projective model structure on R Mod-forms a model category can be found in [Hov99, Section 2.3].…”

Section: The Case Of Chain Complexes In An Abelian Categorymentioning

confidence: 99%

“…Section 6 is devoted to some examples of categories of fibrant objects and their weak cofibrations. Finally, in Section 7, we describe the weak fibrations and weak cofibrations that arise when considering model structures on the category of chain complexes in an abelian category, which were recently introduced by Christensen and Hovey [CH02].…”

Section: Introductionmentioning

confidence: 99%

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Weak (co)fibrations in categories of (co)fibrant objects

Kieboom

Sonck

Linden

et al. 2003

Homology, Homotopy and Applications

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We introduce a fibre homotopy relation for maps in a category of cofibrant objects equipped with a choice of cylinder objects. Weak fibrations are defined to be those morphisms having the weak right lifting property with respect to weak equivalences. We prove a version of Dold's fibre homotopy equivalence theorem and give a number of examples of weak fibrations. If the category of cofibrant objects comes from a model category, we compare fibrations and weak fibrations, and we compare our fibre homotopy relation, which is defined in terms of left homotopies and cylinders, with the fibre homotopy relation defined in terms of right homotopies and path objects. We also dualize our notion of weak fibration in a category of cofibrant objects to a notion of weak cofibration in a category of fibrant objects, and give examples of these weak cofibrations. A section is devoted to the case of chain complexes in an abelian category. IntroductionThe fibre homotopy equivalence theorem of Dold [Dol63, Theorem 6.1] in Top has been generalized by various authors. Besides the original work by Dold, the book [DKP70] of tom Dieck-Kamps-Puppe gives an exposition on weak fibrations (h-Faserungen in Top). Some of the generalizations consider maps which are simultaneously over a given space and under a given space. Booth [Boo93] also obtains versions of Dold's theorem, using suitably defined generalizations of the covering homotopy property. In other cases the fibre homotopy equivalences were studied in a categorical setting, as for example in the Homology, Homotopy and Applications, vol. 5(1), 2003 346 the basic assumption is that the category has some cylinder functor. In the article [Kam72], Kamps uses cylinder functors to define a notion of weak fibration. A model category structure, a concept due to Quillen [Qui67], is another way of introducing a homotopy relation in a category. In fact in a model category there are two dual ways of defining homotopy of maps: left homotopies, defined in terms of cylinder objects, and right homotopies, defined in terms of cocylinder objects. These two methods feature in categories of cofibrant objects and, respectively, categories of fibrant objects. Of these two notions, the latter was introduced by K. S. Brown [Bro73] in 1973 and dualized into the former by Kamps and Porter (see [KP97]). We consider a notion of weak fibration in the context of a category of cofibrant objects with a cylinder object choice, i.e., a chosen cylinder object for every object of the category. Our weak fibrations, and their properties, depend on this cylinder object choice. In case this choice comes from a cylinder functor satisfying certain Kan filler conditions, our fibre homotopy relation coincides with the one used in [KP97]. This makes it possible to compare our weak fibrations with Kamps's.The aim of this article is to study fibre homotopies and weak fibrations in a category of cofibrant objects and, dually, relative homotopies and weak cofibrations in a category of fibrant objects. The presentation is a...

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Section: The Case Of Chain Complexes In An Abelian Categorymentioning

confidence: 99%

Section: The Case Of Chain Complexes In An Abelian Categorymentioning

confidence: 99%

“…Theorem 2.2 of [CH02] gives hypotheses for the classes of P-fibrations, Pcofibrations and P-equivalences to form a model structure…”

Section: The Case Of Chain Complexes In An Abelian Categorymentioning

confidence: 99%

Section: The Case Of Chain Complexes In An Abelian Categorymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Weak (co)fibrations in categories of (co)fibrant objects

Kieboom

Sonck

Linden

et al. 2003

Homology, Homotopy and Applications

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Made-to-Order Weak Factorization Systems

Riehl

2015

Trends in Mathematics

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Localization with respect to a class of maps I — Equivariant localization of diagrams of spaces

Chorny

2005

Isr. J. Math.

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Abstract. Homotopical localizations with respect to a set of maps are known to exist in cofibrantly generated model categories (satisfying additional assumptions) [3,12,20,29]. In this paper we expand the existing framework, so that it will apply to not necessarily cofibrantly generated model categories and, more important, will allow for a localization with respect to a class of maps (satisfying some restrictive conditions).We illustrate our technique by applying it to the equivariant model category of diagrams of spaces [11]. This model category is not cofibrantly generated [7]. We give conditions on a class of maps which ensure the existence of the localization functor; these conditions are satisfied by any set of maps and by the class of maps which induces ordinary localizations on the generalized fixed-points sets.

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Quillen model structures for relative homological algebra

Cited by 71 publications

References 14 publications

Weak (co)fibrations in categories of (co)fibrant objects

Weak (co)fibrations in categories of (co)fibrant objects

Made-to-Order Weak Factorization Systems

Localization with respect to a class of maps I — Equivariant localization of diagrams of spaces

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