model category, model $\infty$-category
on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
The Grothendieck construction may be lifted from categories to model categories, where it serves as a presentation for the (infinity,1)-Grothendieck construction.
Let $M$ be a model category and $F:M\to ModelCat$ a pseudofunctor, where $ModelCat$ is the 2-category of model categories, Quillen adjunctions pointing in the direction of their left adjoints, and mate-pairs of natural isomorphisms. Assume furthermore that:
$F$ is relative, meaning that whenever $f:A\to B$ is a weak equivalence in $M$, then the Quillen adjunction $f_! : F(A) \rightleftarrows F(B) : f^*$ is a Quillen equivalence. (That is, $F$ is a relative functor?.)
$F$ is proper, meaning that whenever $f:A\to B$ is an acyclic cofibration (resp. an acyclic fibration) in $M$, then $f_!$ (resp. $f^*$) preserves weak equivalences.
On the Grothendieck construction $\int F$ we define a morphism $(f,\phi):(A,X) \to (B,Y)$, where $f:A\to B$ in $M$ and $\phi:f_!(X) \to Y$ in $F(B)$, to be:
a weak equivalence if $f:A\to B$ is a weak equivalence in $M$ and $f_!(Q X) \to f_!(X) \xrightarrow{\phi} Y$ is a weak equivalence in $F(B)$, where $Q$ is a cofibrant replacement. Since $f_!\dashv f^*$ is a Quillen equivalencen by relativeness, this is equivalent to the dual condition that $X \xrightarrow{\hat{\phi}} f^*(Y) \to f^*(R Y)$ is a weak equivalence in $F(A)$.
a cofibration if $f$ is a cofibration in $M$ and $\phi : f_!(X)\to Y$ is a cofibration in $F(B)$.
a fibration if $f$ is a fibration in $M$ and the adjunct $\hat{\phi} : X\to f^*(Y)$ is a fibration in $F(A)$.
Then these classes of maps make $\int F$ a model category.
Given a proper relative $F:M\to ModelCat$, we can compose with the underlying $(\infty,1)$-category functor $Ho:ModelCat \to QCat$ with values in (say) quasicategories. Since $F$ is relative, this map takes weak equivalences in $M$ to equivalences of quasicategories, so it induces a functor of quasicategories $Ho(M) \to Ho(QCat) = (\infty,1)Cat$. The (∞,1)-Grothendieck construction of this functor is then equivalent, over $Ho(M)$, to the underlying $(\infty,1)$-category of the Grothendieck-construction model structure on $\int F$; this is Harpaz-Prasma, Proposition 3.1.2.
The first model category version of the Grothendieck construction was given in
This article (Roig 94) had a mistake, which was fixed in
The construction was then generalized in
Another approach is found in
For the special case of pseudofunctors with values in groupoids, a model category version of the Grothendieck construction was discussed in