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
The model structure for right fibrations $(SSet/S)_{rfib}$ is a model category structure on the overcategory $SSet/S$ of simplicial sets over a given quasi-category $S$, that presents the (∞,1)-category of right Kan fibrations of quasi-categories over $S$
This is the $(\infty,1)$-analog of the $(2,1)$-category $Fib_{grpd}(S)$ of categories fibered in groupoids over a category $S$.
Similarly there is an analogous model structure for left fibrations that models left Kan fibrations, i.e. op-fibrations in ∞-groupoids
The extension of this from right fibrations to Cartesian fibrations and from left fibrations to coCartesian fibrations is the model structure for coCartesian fibrations.
The (∞,1)-Grothendieck construction relates this to the global model structure on functors that presents the (∞,1)-category of (∞,1)-functors $Fun(S,\infty Grpd)$ (for left fibrations) and $Fun(S^{op},\infty Grpd)$ (for right fibrations).
The following model category structure is best understood with the (∞,1)-Grothendieck construction in mind, which it serves to model.
Recall from the discussion there that given a morphism $p : X \to S$ of quasi-categories, the (∞,1)-functor $S^{op} \to \infty Grpd$ that the left adjoint to the Grothendieck construction extracts from it is all encoded in the pushout $X^{\triangleleft} \coprod_X S$ in
where $X^\triangleleft = (*) \star X$ is the join of $X$ with the point, i.e. $X$ with an initial object freely adjoined to it.
More discussion of why this is the case is at Adjoints to the Grothendieck construction.
The model category structure described below declares that a morphism in the overcategory $sSet/S$ is a weak equivalence if it induces a weak equivalence of the quasi-categories given by these pushouts. So this is effectively saying that we regard a morphism of right fibration of quasi-categories as a weak equivalences, if under the left adjoint to the $(\infty,1)$-Grothendieck construction it induces a weak equivalences of the $(\infty,1)$-functors that classify these fibrations.
For $f : X \to S$ a morphism of simplicial sets, write $C^{\triangleleft}(f)$ for the pushout
in the category sSet of simplicial sets. Call this the left cone over $f$.
The colimits in sSet are computed componentwise, so that the set of vertices $C^{\triangleleft}(f)_0$ is the disjoint union of the vertices of $S$ and one extra vertex $v$, the cone point.
The model structure for left fibrations or covariant model structure $(sSet/S)_{lfib}$ on $SSet/S$ is given by
A morphism $f : X \to Y$ is
a cofibration if the underlying morphism of simplicial sets is a cofibration in the standard model structure on simplicial sets, i.e. a monomorphism;
a weak equivalence if the induced morphism of cones
is a weak equivalence in the Joyal model structure for quasi-categories, where $X^{\triangleleft}$ is the join $X^{\triangleleft} := {*} \star X$.
This is a
We have
The fibrant objects are precisely the left fibrations (HTT, corollary 2.2.3.12)
Every left anodyne morphism is an acyclic cofibration. (HTT, prop 2.1.4.9)
An alternative characterization of this model structure is:
The model structure for left fibrations is the left Bousfield localization of the overcategory model structure on $SSet/X$ induced by the model structure for quasicategories on $SSet$ at the set of maps $\{ \Delta^n_0 \hookrightarrow \Delta^n | n\ge 0, \Delta^n \to X \}$ indexed by all the simplices of $X$.
This is mentioned in Heuts-Moerdijk, p.5; see also this discussion.
Let $S$ be any simplicial set. Every morphism
in $sSet_S$ for which $X\to Y$ is left anodyne is a weak equivalence in the model structure for left fibrations.
This is HTT, prop. 2.1.4.6.
Recall from here that left anodyne morphisms are the weakly saturated class generated by the horn inclusions (i.e. under transfinite composition of retracts of pushouts). Therefore it is sufficient to check the statement for these generating morphisms.
By the definition of weak equivalences above, this means that we need to check that
is a weak equivalence in $sSet_{Quillen}$.
Observe that this is a pushout
of the inner anodyne morphism $\Lambda[n+1]_{i+1} \to \Delta[n+1]$ and therefore a weak equivalence.
To illustrate the above pushout property set $n = 2$ for example. Start with a 2-simplex $\sigma$ in $S$. Then $(\Delta^2)^{\triangleleft} \coprod_{\Delta^2} S$ is the original simplicial set $S$ together with a tetrahedron $\Delta^3$ built over $\sigma$. One face of the tetrahedron is the original 2-simplex $\sigma$ in $S$, the three others “stick out” of $S$:
The simplicial set $(\Lambda^2_1)^{\triangleleft} \coprod_{\Lambda^2_1} S$ is accordingly the simplicial set $S$ with only two of the three faces of this tetrahedron over $\sigma$ erected.
The map $(\Lambda^3_2) \to (\Delta^2)^{\triangleleft} \coprod_{\Delta^2} S$ identifies the horn of this tetrahedron given by these two new faces and the original face $\sigma$.
The pushout therefore glues in the remaining face of the tetrahedron and fills it with a 3-cell.
For every morphism $j : S \to S'$ we have the corresponding adjunction on overcategories
where
$j_!$ is given by postcomposition of $j$;
$j^*$ is given by pullback along $j$.
(change of base)
This is a Quillen adjunction with respect to the model structures for left fibrations over $S$ and $S'$, respectively. (HTT, prop. 2.1.4.10)
If $j$ is a weak equivalence in $sSet_{Joyal}$, then this is a Quillen equivalence. (HTT, remark. 2.1.4.11)
($(\infty,1)$-Grothendieck construction)
For $C$ a simplicially enriched category and $S = N(C)$ its homotopy coherent nerve, there is a Quillen equivalence
between the model structure for left fibrations over $S$ and the global model structure on sSet-functors on $C$ with values in sSet equipped with the standard model structure on simplicial sets.
For more on this see (∞,1)-Grothendieck construction.
The operadic generalization is the
This is the content of section 2.1.4 of
There the model structure $(sSet/S)_{lfib}$ is called the covariant model structure and the model structure $(sSet/S)_{rfib}$ the contravariant model structure.
The alternative construction as a localization is mentioned in
Further discussion is in