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**On left and right model categories and left and right Bousfield localizations.**
*(English)*
Zbl 1243.18025

In many situations, one would like to construct from an existing model category \(\mathcal {M}\) a new model category, \(L_H\mathcal {M}\) or \(R_H\mathcal {M}\), which has additional weak equivalences from a specified set \(H\) of maps closed under the homotopy relation. This new model category should have the same underlying category as \(\mathcal {M}\) and an appropriate universal property. A left Bousfield localization consists of a left Quillen functor \(\mathcal {M} \to L_H\mathcal {M}\) which is initial among left Quillen functors \(F: \mathcal {M} \to \mathcal {N}\) such that \(\mathbf L F\) maps all elements of \(H\) to isomorphisms in \(\text{Ho}\,\mathcal {N}\). A right Bousfield localization consists of a right Quillen functor \(\mathcal {M} \to R_H\mathcal {M}\) which is initial among right Quillen functors \(U: \mathcal {M} \to \mathcal {N}\) such that \(\mathbf RU\) maps all elements of \(H\) to isomorphisms in \(\text{Ho}\,\mathcal {N}\). In a left Bousfield localization, the cofibrations stay the same, whereas in a right Bousfield localization, the fibrations stay the same.

Neither left nor right Bousfield localizations exist in general. If \(\mathcal {M}\) is left proper and combinatorial, and \(H\) is appropriately small, then a left Bousfield localization \(L_H\mathcal {M}\) exists and is left proper and combinatorial by a theorem J. Smith [Combinatorial model categories, in progress]. The author of the present paper recalls this theorem with proof in Section 4, and emphasizes that the result is solely due to Smith. Usually \(L_H\mathcal {M}\) is not right proper, even if \(\mathcal {M}\) is right proper in addition to the aforementioned hypotheses.

Right Bousfield localizations, when they exist, are usually done with respect to a set \(K\) of isomorphism classes of objects of \(\text{Ho}\,\mathcal {M}\). A morphism \(A \to B\) is a \(K\)-colocal equivalence if for every representative \(X\) of every element of \(K\), the map of homotopy function complexes \(\mathbf R\text{Mor}_\mathcal {M}(X,A) \to \mathbf R\text{Mor}_\mathcal {M}(X,B)\) is a weak equivalence of simplicial sets. A right Bousfield localization \(R_K\mathcal {M}\) of \(\mathcal {M}\) with respect to \(K\) is a right Bousfield localization of \(\mathcal {M}\) with respect to the set of \(K\)-colocal equivalences. By a result of Hirschhorn, if \(\mathcal {M}\) is right proper and cellular, then a right Bousfield localization \(R_K\mathcal {M}\) exists [P.S. Hirschhorn, Model categories and their localizations. Math. Surv. Monogr. 99. Providence: AMS (2003; Zbl 1017.55001)].

Right properness is a strong hypothesis on a model category. One of the main results of the present paper is to remove the right properness hypothesis for the existence of right Bousfield localizations by working with a more general notion of model category called “right model category”. This notion was invented by M. Hovey [Monoidal model categories, arxiv:math/9803002], and is also referred to as “right semi-model category”. This notion was used in M. Spitzweck [Operads, algebras and modules in model categories and motives. Bonn: Univ. Bonn. Mathematisch-Naturwissenschaftliche Fakultät (Dissertation). (2001; Zbl 1103.18300)] and other works. In Section 5, the author proves that if \(\mathcal {C}\) is a tractable model category, then \(R_K \mathcal {C}\) exists as a tractable right \(\mathcal {C}\)-model category.

A right \(\mathcal {C}\)-model category consists of an adjunction \(F: \mathcal {D} \rightleftarrows \mathcal {C} :U\) in which both \(\mathcal {D}\) and \(\mathcal {C}\) are complete and cocomplete categories, each equipped with a class of weak equivalences satisfying the 6-for-2 property, a class of cofibrations closed under retracts and pushouts by arbitrary morphisms, and a class of fibrations closed under retracts and pullbacks by arbitrary morphisms. The axioms for a right \(\mathcal {C}\)-model category require, among other things, that \(F\) preserves cofibrations and trivial cofibrations, and that versions of the usual lifting conditions and functorial factorizations hold in \(\mathcal {D}\) when certain codomains have fibrant \(F\)-images.

The author creates in Section 4 a “collection of techniques for coping with the fact that many important combinatorial model categories are simply not right proper”.

Sections 1, 2, and 3 of the paper recall much of the background on left and right model categories, combinatorial model categories, and tractable model categories. The left/right model category analogues of some Reedy theory are also developed, as well as several inheritance results.

The article also contains a number of applications, some of them are indicated here. Examples of left Bousfield localization are Dugger’s theorem on presentations of combinatorial model categories, homotopy images, and homotopy limits of right Quillen presheaves (these are presheaves of model categories connected by right Quillen functors). Enriched left Bousfield localization, also treated by the author, is applied to describe local model structures on categories of presheaves with values in a symmetric monoidal model category. Applications of right Bousfield localization are homotopy limits of left Quillen presheaves, a Postnikov right model structure, and a stable Postnikov right model structure.

Neither left nor right Bousfield localizations exist in general. If \(\mathcal {M}\) is left proper and combinatorial, and \(H\) is appropriately small, then a left Bousfield localization \(L_H\mathcal {M}\) exists and is left proper and combinatorial by a theorem J. Smith [Combinatorial model categories, in progress]. The author of the present paper recalls this theorem with proof in Section 4, and emphasizes that the result is solely due to Smith. Usually \(L_H\mathcal {M}\) is not right proper, even if \(\mathcal {M}\) is right proper in addition to the aforementioned hypotheses.

Right Bousfield localizations, when they exist, are usually done with respect to a set \(K\) of isomorphism classes of objects of \(\text{Ho}\,\mathcal {M}\). A morphism \(A \to B\) is a \(K\)-colocal equivalence if for every representative \(X\) of every element of \(K\), the map of homotopy function complexes \(\mathbf R\text{Mor}_\mathcal {M}(X,A) \to \mathbf R\text{Mor}_\mathcal {M}(X,B)\) is a weak equivalence of simplicial sets. A right Bousfield localization \(R_K\mathcal {M}\) of \(\mathcal {M}\) with respect to \(K\) is a right Bousfield localization of \(\mathcal {M}\) with respect to the set of \(K\)-colocal equivalences. By a result of Hirschhorn, if \(\mathcal {M}\) is right proper and cellular, then a right Bousfield localization \(R_K\mathcal {M}\) exists [P.S. Hirschhorn, Model categories and their localizations. Math. Surv. Monogr. 99. Providence: AMS (2003; Zbl 1017.55001)].

Right properness is a strong hypothesis on a model category. One of the main results of the present paper is to remove the right properness hypothesis for the existence of right Bousfield localizations by working with a more general notion of model category called “right model category”. This notion was invented by M. Hovey [Monoidal model categories, arxiv:math/9803002], and is also referred to as “right semi-model category”. This notion was used in M. Spitzweck [Operads, algebras and modules in model categories and motives. Bonn: Univ. Bonn. Mathematisch-Naturwissenschaftliche Fakultät (Dissertation). (2001; Zbl 1103.18300)] and other works. In Section 5, the author proves that if \(\mathcal {C}\) is a tractable model category, then \(R_K \mathcal {C}\) exists as a tractable right \(\mathcal {C}\)-model category.

A right \(\mathcal {C}\)-model category consists of an adjunction \(F: \mathcal {D} \rightleftarrows \mathcal {C} :U\) in which both \(\mathcal {D}\) and \(\mathcal {C}\) are complete and cocomplete categories, each equipped with a class of weak equivalences satisfying the 6-for-2 property, a class of cofibrations closed under retracts and pushouts by arbitrary morphisms, and a class of fibrations closed under retracts and pullbacks by arbitrary morphisms. The axioms for a right \(\mathcal {C}\)-model category require, among other things, that \(F\) preserves cofibrations and trivial cofibrations, and that versions of the usual lifting conditions and functorial factorizations hold in \(\mathcal {D}\) when certain codomains have fibrant \(F\)-images.

The author creates in Section 4 a “collection of techniques for coping with the fact that many important combinatorial model categories are simply not right proper”.

Sections 1, 2, and 3 of the paper recall much of the background on left and right model categories, combinatorial model categories, and tractable model categories. The left/right model category analogues of some Reedy theory are also developed, as well as several inheritance results.

The article also contains a number of applications, some of them are indicated here. Examples of left Bousfield localization are Dugger’s theorem on presentations of combinatorial model categories, homotopy images, and homotopy limits of right Quillen presheaves (these are presheaves of model categories connected by right Quillen functors). Enriched left Bousfield localization, also treated by the author, is applied to describe local model structures on categories of presheaves with values in a symmetric monoidal model category. Applications of right Bousfield localization are homotopy limits of left Quillen presheaves, a Postnikov right model structure, and a stable Postnikov right model structure.

Reviewer: Thomas Fiore (Dearborn)

### MSC:

18G55 | Nonabelian homotopical algebra (MSC2010) |

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |