related by the 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 . Every object is cofibrant and the fibrant objects of are precisely the Cartesian fibrations over .
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.
A marked simplicial set is
a pair consisting of
and a subset of edges of , called the marked edges,
A morphism of marked simplicial sets is a morphism of simplicial sets that carries marked edges to marked edges in that .
The category of marked simplicial sets is denoted .
for a simplicial set let
or be the minimally marked simplicial set: only the degenerate edges are marked;
or be the maximally marked simplicial set: every edge is marked.
Simple as the above definition is, for seeing some of its properties it is useful to think of in a more abstract way.
Equip this category with a coverage whose only non-trivial covering family is .
A presheaf is separated precisely if the morphism
is a monomorphism, hence if is a subset of . By functoriality this subset contains all the degenerate 1-cells
Therefore we may naturally identify as a simplicial set equipped with a subset of that contains all degenerate 1-cells.
Moreover, a morphism of separated preseheaves on is by definition just a natural transformation between them, which means it is under this interpretation precisely a morphism of simplicial sets that respects the marked 1-simplices.
Notice that is a genuine quasi-topos:
is not a topos.
The category is a cartesian closed category.
By the general logic of the closed monoidal structure on presheaves we have that is cartesian closed. It remains to check that if are marked simplicial sets in that is a monomorphism and similarly for , that then also has this property.
We find that the marked edges of are
and the morphism sends to
Now, by construction, every non-identity morphism in factors through , which implies that if the components of and coincide on , then already the components of and on coincided. By assumption on the values of and on are already fixed, due to the inclusion . Hence is injective, and so formed in is itself a marked simplicial set.
These mapping complexes are characterized by the fact that we have natural bijections
for and . In particular
In words we have
The -simplices of the internal hom are simplicial maps such that when you restrict to (where is the set of marked edges of ), this morphism factors through the marked edges of .
The marked edges of are those simplicial maps such that the restriction of to factors though the marked edges of . In the presence of the previous condition, this says that when you apply the homotopy to a marked edge of paired with the identity at , the result should be marked.
We generalize all this notation from to the overcategory for any given (plain) simplicial set , by declaring
to be the subcomplexes spanned by the cells that respect that map to the base .
This is HTT, remark 188.8.131.52.
Similarly, the -cells of are morphisms over . Again for fixed , maps into a fiber of , but now only hitting Cartesian edges there. But (as discussed at Cartesian morphism), an edge over a point is Cartesian precisely if it is an equivalence.
We have a sequence of adjoint functors
Observe that weak equivalences in the model structure for quasi-categories may be characterized as follows.
This is HTT, lemma 184.108.40.206.
This may be taken as motivation for the following definition.
For every Cartesian fibration , we have that
is a quasi-category and
is the maximal Kan complex inside it.
A morphism in is a Cartesian equivalence if for every Cartesian fibration we have
be a morphism in such that both vertical maps to are Cartesian fibrations. Then the following are equivalent:
This is HTT, lemma 220.127.116.11.
The model structure on marked simplicial over-sets over – also called the Cartesian model structure since it models Cartesian fibrations – is defined as follows.
(Cartesian model structure on )
A morphism in of marked simplicial sets is
The model structure is proposition 18.104.22.168 in HTT. The simplicial enrichment is corollary 22.214.171.124.
Notice that trivially every object in this model structure is cofibrant. The following proposition shows that the above model structure indeed presents the -category of Cartesian fibrations.
This is HTT, prop. 126.96.36.199.
In particular, the fibrant objects of are precisely the quasicategories in which the marked edges are precisely the equivalences. Note that the Cartesian model structure on is not the model structure on an over category induced on from the Cartesian model structure on !
(coCartesian model structure on )
There is another such model structure, with Cartesian fibrations replaced everywhere by coCartesian fibrations.
A class of morphisms with left lifting property again some class of fibrations is usually called anodyne . For instance a left/right/inner anodyne morphism of simplicial sets is one that has the left lifting property against all left/right fibrations or inner fibrations, respectively.
The class of marked anodyne morphisms in as defined in the following is something that comes close to having the left lifting property against all Cartesian fibrations. It does not quite, but is still useful for various purposes.
The collection of marked anodyne morphisms in is the class of morphisms where the generating set consists of
The crucial property of marked anodyne morphisms is the following characterization of morphisms that have the right lifting property with respect to them.
A morphism in has the right lifting property with respect to the class of marked anodyne maps precisely if
This is HTT, prop. 188.8.131.52
Thus, if is a morphism in with RLP against marked anodyne morphisms, then its underlying morphism in is almost a Cartesian fibration: it may fail to be such only due to missing markings in .
However, if all morphisms in are marked, then has the RLP against marked anodyne morphisms precisely when the underlying morphism is a Cartesian fibration and exactly the Cartesian morphisms are marked in , — in other words, precisely if it is a fibrant object in the model structure on .
See also HTT, remark. 184.108.40.206.
(stability under smash product with cofibrations)
Marked anodyne morphisms are stable under “smash product” with cofibrations:
for marked anodyne, and a cofibration, the induced morphism
out of the pushout in is marked anodyne.
This is HTT, prop. 220.127.116.11.
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.
Marked simplicial sets are introduced in section 3.1 of
The model structure on marked simplicial oversets is described in section 3.1.3