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 over a given simplicial set is a presentation for the (∞,1)-category of cartesian fibrations over . The fibrant objects of are the Cartesian fibrations over and the marked edges in the fibrant marked simplicial sets are the Cartesian morphisms.
Notably for this is a presentation of the (∞,1)-category of (∞,1)-categories: as a plain model category this is Quillen equivalent to the model structure for quasi-categories, but it is indeed an -enriched model category (i.e. enriched over the ordinary model structure on simplicial sets that models ∞-groupoids).
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 , defined by the properties
When instead using as hom-objects
then one obtains an -enriched model category (enriched over the model structure for quasi-categories). This models the full (∞,2)-category of cartesian fibrations of (∞,1)-categories.
For and a cartesian fibration we have
is a quasi-category
is the largest Kan complex in
The model structure on marked simplicial over-sets over – also called the Cartesian model structure since it models cartesian fibrations – is defined as follows.
(Cartesial model structure on )
The category of marked simplicial sets over a marked simplicial set carries a structure of a proper combinatorial simplicial model category defined as follows.
The SSet-enrichment is given by
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 equivalence, namely such that for every cartesian fibration we have that
the induced morphism is an equivalence of quasi-categories
or equivalently: the induced morphism is a equivalence of Kan complexes.
a fibration if it has the right lifting property with respect to all acyclic cofibrations.
(Recall that is the maximal Kan complex in .)
The model structure is proposition 3.1.3.7 in HTT. The simplicial enrichment is corollary 3.1.4.4.
(coCartesial model structure on )
There is another such model structure, with Cartesian fibrations replaced everywhere by coCartesian fibrations.
The Joyal model structure for quasi-categories is an enriched category enriched over itself. So it is not a simplicial model category in the standard sense, which means -enriched.
Indeed, the full sSet-enriched subcategory on fibrant-cofibrant objects is a model for the (∞,2)-category (∞,1)Cat of (∞,1)-categories. For many applications it is more convenient to work just with the (∞,1)-category of (∞,1)-categories inside that, obtained by taking in each hom-object quasi-category the maximal Kan complex.
The resulting (∞,1)-category should have a presentation by a simplicial model category. And the model structure on marked simplicial sets does accomplish this.
The collection of marked anodyne morphisms in is the class of morphisms where the generating set consists of
for the minimally marked horn inclusions
for the horn inclusion with the last edge markeed:
where is the union of all degenete edges in together with the edge .
the inclusion
for every Kan complex the morphism .
A morphism in has the right lifting property with respect to the class of marked anodyne maps precisely if
is an inner fibration
an edge of is marked precisely if it is a Cartesian morphism and is marked in
for every object of and every marked edge in there exists a marked edge of with .
Every morphism in whose underlying morphism in is marked anodyne is a Cartesian equivalence.
section 3.1.3 of