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 .
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 .
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
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