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simplicial Quillen adjunction

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Model category theory

model category

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Enriched category theory

Contents

Definition

A simplicial Quillen adjunction is an sSet-enriched Quillen adjunction: an adjunction

(LR):CD (L \dashv R) : C \stackrel{\leftarrow}{\to} D

of sSet-enriched functors between simplicial model categories CC and DD, such that the underlying adjunction of ordinary functors is a Quillen adjunction between the model category structures underlying the simplicial model categories.

Properties

Presentation of \infty-adjunctions

Simplicial Quillen adjunctions model pairs of adjoint (∞,1)-functors in a fairly immediate manner: their restriction to fibrant-cofibrant objects is the sSet-enriched functor that presents the (,1)(\infty,1)-derived functor under the model of (∞,1)-categories by simplicially enriched categories.

Proposition

Let CC and DD be simplicial model categories and let

(LR):CRLD (L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D

be an sSet-enriched adjunction whose underlying ordinary adjunction is a Quillen adjunction. Let C C^\circ and D D^\circ be the (∞,1)-categories presented by CC and DD (the Kan complex-enriched full sSet-subcategories on fibrant-cofibrant objects). Then the Quillen adjunction lifts to a pair of adjoint (∞,1)-functors

(𝕃):C D . (\mathbb{L} \dashv \mathbb{R}) : C^\circ \stackrel{\leftarrow}{\to} D^{\circ} \,.

On the decategorified level of the homotopoy categories these are the total left and right derived functors, respectively, of LL and RR.

Proof

This is proposition 5.2.4.6 in HTT.

Recognition

The following proposition states conditions under which a simplicial Quillen adjunction may be detected already from knowing of the right adjoint only that it preserves fibrant objects (instead of all fibrations).

Proposition

If CC and DD are simplicial model categories and DD is a left proper model category, then for an sSet-enriched adjunction

(LR):CD (L \dashv R) : C \stackrel{\leftarrow}{\to} D

to be a Quillen adjunction it is already sufficient that LL preserves cofibrations and RR just fibrant objects.

This appears as HTT, cor. A.3.7.2.

Remark

This is in particular useful for finding simplicial Quillen adjunctions into left Bousfield localizations of left proper model categories: the left Bousfield localization keeps the cofibrations unchanged and preserves left properness, and the fibrant objects in the Bousfield localized structure have a good characterization: they are the fibrant objects in the original model structure that are also local objects with respect to the set of morphisms at which one localizes.

Therefore for DD the left Bousfield localization of a simplicial left proper model category EE at a class SS of morphisms, for checking the Quillen adjunction property of (LR)(L \dashv R) it is sufficient to check that LL preserves cofibrations, and that RR takes fibrant objects cc of CC to such fibrant objects of EE that have the property that for all fSf \in S the derived hom-space map Hom(f,R(c))\mathbb{R}Hom(f,R(c)) is a weak equivalence.

Revised on April 18, 2012 05:15:16 by Urs Schreiber (82.169.65.155)