nLab
model structure on marked simplicial over-sets

Idea

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

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 ${\mathrm{Set}}_S^{+}$ 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 $S$.

Remark

For $X\in {\mathrm{SSet}}^{+}/S$ abd $p:Y\to X$ 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.

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 fibration, 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 homotopy 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 ${\mathrm{Map}}_S^{♭}(X,Z^{\mathrm{cart}})$.)

Proposition

This defines a proper combinatorial simplicial model category.

Proof

This is proposition 3.1.3.7 in HTT.

References

section 3.1.3 of

• Jacob Lurie, Higher Topos Theory