nLab
model structure on marked simplicial over-sets

Idea

The model category structure on the category ${\mathrm{SSet}}^{+}/S$ of marked simplicial sets over a given simplicial set $S$ is a presentation for the (∞,1)-category of cartesian fibrations over $S$. The fibrant objects of ${\mathrm{SSet}}^{+}/S$ are the Cartesian fibrations over $S$ and the marked edges in the fibrant marked simplicial sets are the Cartesian morphisms.

Notably for $S=*$ 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 ${\mathrm{sSet}}_{\mathrm{Quillen}}$-enriched model category (i.e. enriched over the ordinary model structure on simplicial sets that models ∞-groupoids).

The $(\mathrm{\infty }\mathrm{,1})$-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.

Definition

Marked simplicial over-sets

Let $S$ be a fixed simplicial set. Recall from the notation at marked simplicial set that $S^{\#}$ denotes the maximally marked simplicial set of $S$ where all edges are marked edges.

Write ${\mathrm{SSet}}^{+}/S^{\#}$ – or ${\mathrm{SSet}}^{+}/S$ for short – for the over category of the category ${\mathrm{Set}}^{+}$ of marked simplicial sets over $S^{\#}$.

Recall the notation from marked simplicial set. For $X$ and $Y$ in ${\mathrm{SSet}}_S^{+}$ write ${\mathrm{Map}}_S^{♭}(X,Y)\subset {\mathrm{Map}}^{♭}(X,Y)$ and ${\mathrm{Map}}_S^{\#}(X,Y)\subset {\mathrm{Map}}^{\#}(X,Y)$ for the simplicial subsets of maps compatible with the maps to $S$, defined by the properties

$${\mathrm{Hom}}_{\mathrm{SSet}}(K,{\mathrm{Map}}^{♭}(X,Y))\simeq {\mathrm{Hom}}_{{\mathrm{SSet}}^{+}}(K^{♭}\times X,Y)$$Hom_{SSet}(K, Map^\flat(X,Y)) \simeq Hom_{SSet^+}(K^\flat \times X, Y)

$${\mathrm{Hom}}_{\mathrm{SSet}}(K,{\mathrm{Map}}^{\#}(X,Y))\simeq {\mathrm{Hom}}_{{\mathrm{SSet}}^{+}}(K^{\#}\times X,Y).$$Hom_{SSet}(K, Map^#(X,Y)) \simeq Hom_{SSet^+}(K^# \times X, Y) \,.

Remark (HTT, 3.1.4.5)

When instead using as hom-objects

$$\mathrm{hom}(X,Y):={\mathrm{Map}}_S^{♭}(X,Y)$$hom(X,Y) := Map_S^\flat(X,Y)

then one obtains an ${\mathrm{sSet}}_{\mathrm{Joyal}}$-enriched model category (enriched over the model structure for quasi-categories). This models the full (∞,2)-category of cartesian fibrations of (∞,1)-categories.

Remark (HTT, 3.1.3.1)

For $(X\to S)\in {\mathrm{SSet}}^{+}/S$ and $p:Y\to S$ a cartesian fibration we have

• ${\mathrm{Map}}_S^{♭}(X,Y^{\mathrm{cart}})$ is a quasi-category

• ${\mathrm{Map}}_S^{\#}(X,Y^{\mathrm{cart}})$ is the largest Kan complex in ${\mathrm{Map}}_S^{♭}(X,Y^{\mathrm{cart}})$

The model structure

The model structure on marked simplicial over-sets ${\mathrm{Set}}^{+}/S$ over $S\in \mathrm{SSet}$ – also called the Cartesian model structure since it models cartesian fibrations – is defined as follows.

Definition/Proposition

(Cartesial model structure on ${\mathrm{sSet}}^{+}/S$)

The category ${\mathrm{SSet}}^{+}/S$ of marked simplicial sets over a marked simplicial set $S$ carries a structure of a proper combinatorial simplicial model category defined as follows.

The SSet-enrichment is given by

$$\mathrm{hom}(X,Y):={\mathrm{Map}}_S^{\#}(X,Y).$$hom(X,Y) := Map_S^#(X,Y) \,.

A morphism $f:X\to X\prime$ in ${\mathrm{SSet}}^{+}/S$ 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 $Z\to S$ we have that

  • the induced morphism ${\mathrm{Map}}_S^{♭}(Y,Z^{\mathrm{cart}})\to {\mathrm{Map}}_S^{♭}(X,Z^{\mathrm{cart}})$ is an equivalence of quasi-categories

  • or equivalently: the induced morphism ${\mathrm{Map}}_S^{\#}(Y,Z^{\mathrm{cart}})\to {\mathrm{Map}}_S^{\#}(X,Z^{\mathrm{cart}})$ is a equivalence of Kan complexes.

• a fibration if it has the right lifting property with respect to all acyclic cofibrations.

(Recall that ${\mathrm{Map}}_S^{\#}(X,Z^{\mathrm{cart}})$ is the maximal Kan complex in ${\mathrm{Map}}_S^{♭}(X,Z^{\mathrm{cart}})$.)

Proof

The model structure is proposition 3.1.3.7 in HTT. The simplicial enrichment is corollary 3.1.4.4.

Definition/Proposition

(coCartesial model structure on ${\mathrm{sSet}}^{+}/S$)

There is another such model structure, with Cartesian fibrations replaced everywhere by coCartesian fibrations.

As a model for the $(\mathrm{\infty }\mathrm{,1})$-category of $(\mathrm{\infty }\mathrm{,1})$-categories

The Joyal model structure for quasi-categories ${\mathrm{sSet}}_{\mathrm{Joyal}}$ is an enriched category enriched over itself. So it is not a simplicial model category in the standard sense, which means ${\mathrm{sSet}}_{\mathrm{Quillen}}$-enriched.

Indeed, the full sSet-enriched subcategory $({\mathrm{sSet}}_{\mathrm{Joyal}}{)}^{\circ }$ 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.

Details

Marked anodyne morphisms

Definition (HTT, Def 3.1.1.1)

The collection of marked anodyne morphisms in ${\mathrm{SSet}}^{+}/S$ is the class of morphisms ${\mathrm{An}}^{+}=\mathrm{LLP}(\mathrm{RLP}({\mathrm{An}}_0^{+}))$ where the generating set ${\mathrm{An}}_0^{+}$ consists of

• for $0<i<n$ the minimally marked horn inclusions

  $$(\Lambda [n{]}_i{)}^{♭}\to \Delta [n{]}^{♭}$$(\Lambda[n]_i)^\flat \to \Delta[n]^\flat

• for $i=n$ the horn inclusion with the last edge markeed:

  $$(\Lambda [n{]}_i,ℰ\cap (\Lambda [n{]}_i{)}_1)\to (\Delta [n],ℰ),$$(\Lambda[n]_i, \mathcal{E} \cap (\Lambda[n]_i)_1) \to (\Delta[n], \mathcal{E} ) \,,

  where $ℰ$ is the union of all degenete edges in $\Delta [n]$ together with the edge ${\Delta }^{\{n-1,n\}}\to \Delta [n]$.

• the inclusion

  $$(\Lambda [2{]}_1{)}^{♯}\coprod _{(\Lambda [2{]}_1{)}^{♭}}(\Delta [2]{)}^{♭}\to (\Delta [2]{)}^{♯}.$$(\Lambda[2]_1)^\sharp \coprod_{(\Lambda[2]_1)^\flat} (\Delta[2])^\flat \to (\Delta[2])^\sharp \,.

• for every Kan complex $K$ the morphism $K^{♭}\to K^{\#}$.

Proposition (HTT, prop. 3.1.1.6)

A morphism $p:X\to S$ in ${\mathrm{SSet}}^{+}$ has the right lifting property with respect to the class ${\mathrm{An}}^{+}$ of marked anodyne maps precisely if

1. $p$ is an inner fibration

2. an edge $e$ of $X$ is marked precisely if it is a Cartesian morphism and $p(e)$ is marked in $S$

3. for every object $y$ of $X$ and every marked edge $\overline{e}:\overline{x}\to p(y)$ in $S$ there exists a marked edge $e:x\to y$ of $X$ with $p(e)=\overline{e}$.

Remark (HTT, prop. 3.1.3.4)

Every morphism $f:X\to Y$ in ${\mathrm{SSet}}^{+}/S$ whose underlying morphism in ${\mathrm{SSet}}^{+}$ is marked anodyne is a Cartesian equivalence.

References

section 3.1.3 of

• Jacob Lurie, Higher Topos Theory