nLab
simplicial Quillen adjunction

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Definition

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

(L⊣R):C→←D(L \dashv R) : C \stackrel{\leftarrow}{\to} D

of sSet-enriched functors between simplicial model categories C and D, 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 ∞-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)-derived functor under the model of (∞,1)-categories by simplicially enriched categories.

Proposition

Let C and D be simplicial model categories and let

(L⊣R):C→R←LD(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 ∘ and D ∘ be the (∞,1)-categories presented by C and D (the Kan complex-enriched full sSet-subcategories on fibrant-cofibrant objects). Then the Quillen adjunction lifts to a pair of adjoint (∞,1)-functors

(𝕃⊣ℝ):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 L and R.

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 C and D are simplicial model categories and D is a left proper model category, then for an sSet-enriched adjunction

(L⊣R):C→←D(L \dashv R) : C \stackrel{\leftarrow}{\to} D

to be a Quillen adjunction it is already sufficient that L preserves cofibrations and R 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 D the left Bousfield localization of a simplicial left proper model category E at a class S of morphisms, for checking the Quillen adjunction property of (L⊣R) it is sufficient to check that L preserves cofibrations, and that R takes fibrant objects c of C to such fibrant objects of E that have the property that for all f∈S the derived hom-space map ℝHom(f,R(c)) is a weak equivalence.

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