model category, model -category
Definitions
Morphisms
Universal constructions
Refinements
Producing new model structures
Presentation of -categories
Model structures
for -groupoids
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant -groupoids
for rational -groupoids
for rational equivariant -groupoids
for -groupoids
for -groups
for -algebras
general -algebras
specific -algebras
for stable/spectrum objects
for -categories
for stable -categories
for -operads
for -categories
for -sheaves / -stacks
A simplicial Quillen adjunction is an sSet-enriched Quillen adjunction: an enriched adjunction
of sSet-enriched functors between simplicial model categories and , such that the underlying adjunction of ordinary functors is a Quillen adjunction between the model category structures underlying the simplicial model categories.
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 -derived functor under the model of (∞,1)-categories by simplicially enriched categories.
Let and be simplicial model categories and let
be an sSet-enriched adjunction whose underlying ordinary adjunction is a Quillen adjunction. Let and be the (∞,1)-categories presented by and (the Kan complex-enriched full sSet-subcategories on fibrant-cofibrant objects). Then the Quillen adjunction lifts to a pair of adjoint (∞,1)-functors
On the decategorified level of the homotopoy categories these are the total left and right derived functors, respectively, of and .
This is proposition 5.2.4.6 in HTT.
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).
(recognition of simplicial Quillen adjunctions)
If
and are simplicial model categories
then for an sSet-enriched adjunction
to be a Quillen adjunction it is already sufficient that
preserves cofibrations
preserves fibrant objects.
This appears as HTT, cor. A.3.7.2.
Prop. is particularly 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 the left Bousfield localization of a simplicial left proper model category at a class of morphisms, for checking the Quillen adjunction property of it is sufficient to check that preserves cofibrations, and that takes fibrant objects of to such fibrant objects of that have the property that for all the derived hom-space map is a weak equivalence.
simplicial Quillen adjunction
On the enhancement of plain Quillen adjunctions between left proper combinatorial model categories to simplicial Quillen adjunctions:
Last revised on July 4, 2022 at 16:39:18. See the history of this page for a list of all contributions to it.