on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
For a model category and an object, the over category as well as the undercategory inherit themselves structures of model categories whose fibrations, cofibrations and weak equivalences are precisely the morphism that become fibrations, cofibrations and weak equivalences in under the forgetful functor or .
If is
then so are and .
More in detail, if are the classes of generating cofibrations and of generating acylic cofibrations of , respectively, then
This is spelled out in (Hirschhorn 05).
If is a combinatorial model category, then so is .
By basic properties of locally presentable categories they are stable under slicing. Hence with locally presentable also is, and by prop. with cofibrantly generated also is.
If is an cartesian enriched model category?, then so is .
By basic properties of cartesian enriched categories? they are stable under slicing, where tensoring is computed in . Hence with enriched also is. The pushout product axiom now follows from the fact that in overcategories pushouts can be computed in the underlying category . The unit axiom? follows from the unit axiom of using the fact that tensorings are computed in .
If is a simplicial model category and is fibrant, then the overcategory with the above slice model structure is a presentation of the over-(∞,1)-category : we have an equivalence of (∞,1)-categories
It is clear that we have an essentially surjective (∞,1)-functor . What has to be shown is that this is a full and faithful (∞,1)-functor in that it is an equivalence on all hom-∞-groupoids .
To see this, notice that the hom-space in an over-(∞,1)-category between objects and is given (as discussed there) by the (∞,1)-pullback
in ∞Grpd.
Let be a cofibrant representative and be a fibration representative in (which always exists) of the objects of these names in , respectively. In terms of these we have a cofibration
in , exhibiting as a cofibrant object of ; and a fibration
in , exhibiting as a fibrant object in .
Moreover, the diagram in sSet given by
is
a pullback diagram in sSet (by the definition of morphism in an ordinary overcategory);
a homotopy pullback in the model structure on simplicial sets, because by the axioms on the sSet enriched model category and the above (co)fibrancy assumptions, all objects are Kan complexes and the right vertical morphism is a Kan fibration.
has in the top left the correct derived hom-space in (since is cofibrant and fibrant).
This means that this correct hom-space is indeed a model for .
Given an adjunction with and , the following compositions define two Quillen ajdunctions between associated slice categories. If , then
is an adjunction, where is the composition , the second arrow is the base change functor along the unit . If , then
is an adjunction, where . The first adjunction is a Quillen equivalence if is cofibrant and is fibrant. The second adjunction is a Quillen equivalence if is fibrant.
These adjunctions are Quillen adjunctions because their left (respectively right) adjoints are left (respectively right) Quillen functors: in the model structures on slice categories (co)fibrations and weak equivalences are created by the forgetful functor to or , see Hirschhorn’s note (Hirschhorn 05). An object in given by an arrow is cofibrant if and only if is cofibrant and fibrant if and only if is a fibration. Quillen’s criterion for Quillen equivalences now yields the statements about equivalences.
model structure on an over-category
Philip Hirschhorn, Theorem 7.6.5 of: Model Categories and Their Localizations, AMS Math. Survey and Monographs Vol 99 (2002) (ISBN:978-0-8218-4917-0, pdf toc, pdf)
Philip Hirschhorn, Overcategories and undercategories of model categories, 2005 (pdf, arXiv:1507.01624)
Last revised on August 29, 2020 at 10:15:10. See the history of this page for a list of all contributions to it.