on dg-algebras/on dg-coalgebras and on on cosimplicial rings (related by monoidal Dold-Kan correspondence)
The model category structure on the category of marked simplicial sets sitting over a given simplicial set is a presentation for the (∞,1)-category of cartesian fibrations over .
The -categorical Grothendieck construction that exhibits the correspondence between cartesian fibrations and (∞,1)-presheaves is in turn modeled by a Quillen equivalence between the model structure on marked simplicial over-sets and the projective global model structure on simplicial presheaves.
Let be a fixed simplicial set. Recall from the notation at marked simplicial set that denotes the maximally marked simplicial set of where all edges are marked edges.
Write – or for short – for the over category of the category of marked simplicial sets over .
Recall the notation from marked simplicial set. For and in write and for the simplicial subsets of maps compatible with the maps to .
The model structure on marked simplicial over-sets over – also called the Cartesian model structure since it models cartesian fibrations – is defined as follows.
A morphism in of marked simplicial sets 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).
(def. 3.1.2.2 of HTT)
a weak equivalence if it is a Cartesian fibration, namely such that for every cartesian fibration we have that
the induced morphism is an equivalence of quasi-categories
a fibration if it has the right lifting property with respect to all acyclic cofibrations.
(Recall that is the maximal Kan complex in .)
This defines a proper combinatorial simplicial model category.
This is proposition 3.1.3.7 in HTT.
section 3.1.3 of