on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
on algebras over an operad, on modules over an algebra over an operad
on dendroidal sets, for dendroidal complete Segal spaces, for dendroidal Cartesian fibrations
For $C$ a model category and $X \in C$ an object, the over category $C/X$ as well as the undercategory $X/C$ inherit themselves structures of model categories whose fibrations, cofibrations and weak equivalences are precisely the morphism that become fibrations, cofibrations and weak equivalences in $C$ under the forgetful functor $C/X \to C$ or $X/C \to C$.
If $\mathcal{C}$ is
then so are $\mathcal{C}_{/X}$ and $\mathcal{C}^{X/}$.
More in detail, if $I,J \subset Mor(\mathcal{C})$ are the classes of generating cofibrations and of generating acylic cofibrations of $\mathcal{C}$, respectively, then
This is spelled out in (Hirschhorn 05).
If $\mathcal{C}$ is a combinatorial model category, then so is $\mathcal{C}_{/X}$.
By basic properties of locally presentable categories they are stable under slicing. Hence with $\mathcal{C}$ locally presentable also $\mathcal{C}_{/X}$ is, and by prop. 2 with $\mathcal{C}$ cofibrantly generated also $\mathcal{C}_{/X}$ is.
If $\mathcal{C}$ is an cartesian enriched model category?, then so is $\mathcal{C}_{/X}$.
By basic properties of cartesian enriched categories? they are stable under slicing, where tensoring is computed in $\mathcal{C}$. Hence with $\mathcal{C}$ enriched also $\mathcal{C}_{/X}$ is. The pushout product axiom now follows from the fact that in overcategories pushouts can be computed in the underlying category $\mathcal{C}$. The unit axiom? follows from the unit axiom of $\mathcal{C}$ using the fact that tensorings are computed in $\mathcal{C}$.
If $C$ is a simplicial model category and $X \in C$ is fibrant, then the overcategory $C/X$ with the above slice model structure is a presentation of the over-(∞,1)-category $C^\circ / X$: we have an equivalence of (∞,1)-categories
It is clear that we have an essentially surjective (∞,1)-functor $C^\circ/X \to (C/X)^\circ$. 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 $C^\circ/X(a,b) \simeq (C/X)^\circ(a,b)$.
To see this, notice that the hom-space in an over-(∞,1)-category $C^\circ/X$ between objects $a : A \to X$ and $b : B \to X$ is given (as discussed there) by the (∞,1)-pullback
in ∞Grpd.
Let $A \in C$ be a cofibrant representative and $b : B \to X$ be a fibration representative in $C$ (which always exists) of the objects of these names in $C^\circ$, respectively. In terms of these we have a cofibration
in $C/X$, exhibiting $a$ as a cofibrant object of $C/X$; and a fibration
in $C/X$, exhibiting $b$ as a fibrant object in $C/X$.
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${}_{Quillen}$ enriched model category $C$ 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 $C/X$ (since $a$ is cofibrant and $b$ fibrant).
This means that this correct hom-space $C/X(a,b) \simeq (C/X)^\circ(a,b) \in sSet$ is indeed a model for $C^\circ/X(a,b) \in \infty Grpd$.
Given an adjunction $L\dashv R$ with $L\colon A\to B$ and $R\colon B\to A$, the following compositions define two Quillen ajdunctions between associated slice categories. If $X\in A$, then
is an adjunction, where is the composition $R\colon B/LX\to A/RLX\to A/X$, the second arrow is the base change functor along the unit $X\to RLX$. If $Y\in B$, then
is an adjunction, where $L\colon A/RY\to B/LRY\to B/Y$. The first adjunction is a Quillen equivalence if $X$ is cofibrant and $LX$ is fibrant. The second adjunction is a Quillen equivalence if $Y$ 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~$A$ or~$B$, see Hirschhorn’s note (Hirschhorn 05). An object in $A/X$ given by an arrow $Z\to X$ is cofibrant if and only if $Z$ is cofibrant and fibrant if and only if $Z\to X$ is a fibration. Quillen’s criterion for Quillen equivalences now yields the statements about equivalences.
model structure on an over-category