# nLab model structure for Cartesian fibrations

model category

## Model structures

for ∞-groupoids

### for $\left(\infty ,1\right)$-sheaves / $\infty$-stacks

#### $\left(\infty ,1\right)$-Category theory

(∞,1)-category theory

# Contents

## 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 $S$. Every object is cofibrant and the fibrant objects of ${\mathrm{SSet}}^{+}/S$ are precisely the Cartesian fibrations over $S$.

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 $\left(\infty ,1\right)$-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.

## Marked simplicial sets

Marked simplicial sets are simplicial sets with a little bit of extra structure: a marking that remembers which edges are supposed to be Cartesian morphisms.

### Definition

#### In components

###### Definition

A marked simplicial set is

• a pair $\left(S,E\right)$ consisting of

• a simplicial set $S$

• and a subset $E\subset {S}_{1}$ of edges of $S$, called the marked edges,

• such that

• all degenerate edges are marked edges.

A morphism $\left(S,E\right)\to \left(S\prime ,E\prime \right)$ of marked simplicial sets is a morphism $f:S\to S\prime$ of simplicial sets that carries marked edges to marked edges in that $f\left(E\right)\subset E\prime$.

###### Notation
• The category of marked simplicial sets is denoted ${\mathrm{sSet}}^{+}$.

• for $S$ a simplicial set let

• ${S}^{♭}$ or ${S}^{\mathrm{min}}$ be the minimally marked simplicial set: only the degenerate edges are marked;

• ${S}^{♯}$ or ${S}^{\mathrm{max}}$ be the maximally marked simplicial set: every edge is marked.

• for $p:X\to S$ a Cartesian fibration of simplicial sets let

• ${X}^{♮}$ or ${X}^{\mathrm{cart}}$ be the cartesian marked simplicial set: precisely the $p$-cartesian morphisms are marked

#### As a quasi-topos

Simple as the above definition is, for seeing some of its properties it is useful to think of ${\mathrm{sSet}}^{+}$ in a more abstract way.

###### Definition

Let ${\Delta }^{+}$ be the category defined as the simplex category $\Delta$, but with one more object $\left[{1}^{+}\right]$ that factors the unique morphism $\left[1\right]\to \left[0\right]$ in $\Delta$

$\begin{array}{ccc}\left[0\right]& \stackrel{\to }{\to }& \left[1\right]\\ {}^{=}↓& ↙& {↓}^{p}\\ \left[0\right]& ←& \left[{1}^{+}\right]\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ [0] &\stackrel{\to}{\to}& [1] \\ {}^{\mathllap{=}}\downarrow &\swarrow& \downarrow^{\mathrlap{p}} \\ [0] &\leftarrow& [1^+] } \,.

Equip this category with a coverage whose only non-trivial covering family is $\left\{p:\left[1\right]\to \left[1{\right]}^{+}\right\}$.

###### Observation

The category ${\mathrm{sSet}}^{+}$ is the quasi-topos of separated presheaves on ${\Delta }^{+}$:

${\mathrm{sSet}}^{+}\simeq \mathrm{SepPSh}\left({\Delta }^{+}\right)\phantom{\rule{thinmathspace}{0ex}}.$sSet^+ \simeq SepPSh(\Delta^+) \,.
###### Proof

A presheaf $X:\left({\Delta }^{+}{\right)}^{\mathrm{op}}\to \mathrm{Set}$ is separated precisely if the morphism

$X\left(p\right):{X}_{{1}^{+}}\to {X}_{1}$X(p) : X_{1^+} \to X_1

is a monomorphism, hence if ${X}_{{1}^{+}}$ is a subset of ${X}_{1}$. By functoriality this subset contains all the degenerate 1-cells

$\begin{array}{c}{X}_{0}\\ ↓& {↘}^{\sigma }\\ {X}_{{1}^{+}}& ↪& {X}_{1}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ X_0 \\ \downarrow & \searrow^{\mathrlap{\sigma}} \\ X_{1^+} &\hookrightarrow& X_1 } \,.

Therefore we may naturally identify $X$ as a simplicial set equipped with a subset of ${X}_{1}$ that contains all degenerate 1-cells.

Moreover, a morphism of separated preseheaves on ${\Delta }^{+}$ 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 ${\mathrm{sSet}}^{+}$ is a genuine quasi-topos:

###### Observation

${\mathrm{sSet}}^{+}$ is not a topos.

###### Proof

The canonical morphisms ${X}^{♭}\to {X}^{♯}$ are monomorphisms and epimorphisms, but not isomorphisms. Therefore ${\mathrm{sSet}}^{+}$ is not a balanced category, hence cannot be a topos.

### Cartesian closure

###### Lemma

The category ${\mathrm{sSet}}^{+}$ is a cartesian closed category.

###### Proof

This is an immediate consequence of the above observation that ${\mathrm{sSet}}^{+}$ is a quasitopos. But it is useful to spell out the Cartesian closure in detail.

By the general logic of the closed monoidal structure on presheaves we have that $\mathrm{PSh}\left({\Delta }^{+}\right)$ is cartesian closed. It remains to check that if $X,Y\in \mathrm{PSh}\left({\Delta }^{+}\right)$ are marked simplicial sets in that ${X}_{{1}^{+}}\to {X}_{1}$ is a monomorphism and similarly for $Y$, that then also ${Y}^{X}$ has this property.

We find that the marked edges of ${Y}^{X}$ are

$\left({Y}^{X}{\right)}_{{1}^{+}}\simeq {\mathrm{Hom}}_{\mathrm{PSh}\left({\Delta }^{+}\right)}\left(\left[{1}^{+}\right],{Y}^{X}\right)\simeq {\mathrm{Hom}}_{\mathrm{PSh}\left({\Delta }^{+}\right)}\left(\left[{1}^{+}\right]×X,Y\right)$(Y^X)_{1^+} \simeq Hom_{PSh(\Delta^+)}([1^+], Y^X) \simeq Hom_{PSh(\Delta^+)}([1^+] \times X, Y)

and the morphism $\left({Y}^{X}{\right)}_{{1}^{+}}\to \left({Y}^{X}{\right)}_{1}$ sends $X×\left[{1}^{+}\right]\stackrel{\eta }{\to }Y$ to

$X×\left[1\right]\stackrel{\left(\mathrm{Id},p\right)}{\to }X×\left[{1}^{+}\right]\stackrel{\eta }{\to }Y\phantom{\rule{thinmathspace}{0ex}}.$X \times [1] \stackrel{(Id,p)}{\to} X \times [1^+] \stackrel{\eta}{\to} Y \,.

Now, by construction, every non-identity morphism $U\to \left[{1}^{+}\right]$ in ${\Delta }^{+}$ factors through $U\to \left[1\right]$, which implies that if the components of ${p}^{*}{\eta }_{1}$ and ${p}^{*}{\eta }_{2}$ coincide on $U\ne \left[{1}^{+}\right]$, then already the components of ${\eta }_{1}$ and ${\eta }_{2}$ on $U$ coincided. By assumption on $X$ the values of ${\eta }_{1}$ and ${\eta }_{2}$ on $U=\left[{1}^{+}\right]$ are already fixed, due to the inclusion ${X}_{{1}^{+}}×\left[{1}^{+}{\right]}_{{1}^{+}}↪{X}_{1}×\left[{1}^{+}{\right]}_{1}$. Hence ${p}^{*}$ is injective, and so ${Y}^{X}$ formed in $\mathrm{PSh}\left({\Delta }^{+}\right)$ is itself a marked simplicial set.

###### Definition
• For $X$ and $Y$ marked simplicial sets let

• ${\mathrm{Map}}^{♭}\left(X,Y\right)$ be the simplicial set underlying the cartesian internal hom ${Y}^{X}\in {\mathrm{sSet}}^{+}$

• ${\mathrm{Map}}^{♯}\left(X,Y\right)$ the simplicial set consisting of all simplices $\sigma \in {\mathrm{Map}}^{♭}\left(X,Y\right)$ such that every edge of $\sigma$ is a marked edge of ${Y}^{X}$.

###### Corollary

These mapping complexes are characterized by the fact that we have natural bijections

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

and

${\mathrm{Hom}}_{\mathrm{sSet}}\left(K,{\mathrm{Map}}^{♯}\left(X,Y\right)\right)\simeq {\mathrm{Hom}}_{{\mathrm{sSet}}^{+}}\left({K}^{♯},{\mathrm{Map}}^{♯}\left(X,Y\right)\right)\simeq {\mathrm{Hom}}_{{\mathrm{sSet}}^{+}}\left({K}^{♯}×X,Y\right)$Hom_{sSet}(K, Map^\sharp(X,Y)) \simeq Hom_{sSet^+}(K^\sharp, Map^\sharp(X,Y)) \simeq Hom_{sSet^+}(K^\sharp \times X, Y)

for $K\in \mathrm{sSet}$ and $X,Y\in {\mathrm{sSet}}^{+}$. In particular

${\mathrm{Map}}^{♭}\left(X,Y{\right)}_{n}={\mathrm{Hom}}_{{\mathrm{sSet}}^{+}}\left(X×\Delta \left[n{\right]}^{♭},Y\right)$Map^\flat(X,Y)_n = Hom_{sSet^+}(X \times \Delta[n]^\flat, Y)

and

${\mathrm{Map}}^{♯}\left(X,Y{\right)}_{n}={\mathrm{Hom}}_{{\mathrm{sSet}}^{+}}\left(X×\Delta \left[n{\right]}^{♯},Y\right)\phantom{\rule{thinmathspace}{0ex}}.$Map^\sharp(X,Y)_n = Hom_{sSet^+}(X \times \Delta[n]^\sharp, Y) \,.

In words we have

• The $n$-simplices of the internal hom ${Y}^{X}$ are simplicial maps $X×\Delta \left[n\right]\to Y$ such that when you restrict ${X}_{1}×\Delta \left[n{\right]}_{1}\to {Y}_{1}$ to $E×\Delta \left[n{\right]}_{0}$ (where $E$ is the set of marked edges of $X$), this morphism factors through the marked edges of $Y$.

• The marked edges of ${Y}^{X}$ are those simplicial maps $X×\Delta \left[1\right]\to Y$ such that the restriction of ${X}_{1}×\Delta \left[1{\right]}_{1}\to {Y}_{1}$ to $E×\Delta \left[1{\right]}_{1}$ factors though the marked edges of $Y$. In the presence of the previous condition, this says that when you apply the homotopy $X×\Delta \left[1\right]\to Y$ to a marked edge of $X$ paired with the identity at $\left[1\right]$, the result should be marked.

###### Definition

We generalize all this notation from ${\mathrm{sSet}}^{+}$ to the overcategory ${\mathrm{sSet}}^{+}/S:={\mathrm{sSet}}^{+}/\left({S}^{♯}\right)$ for any given (plain) simplicial set $S$, by declaring

${\mathrm{Map}}_{S}^{♭}\left(X,Y\right)\subset {\mathrm{Map}}^{♭}\left(X,Y\right)$Map_S^\flat(X,Y) \subset Map^\flat(X,Y)

and

${\mathrm{Map}}_{S}^{♯}\left(X,Y\right)\subset {\mathrm{Map}}^{♯}\left(X,Y\right)$Map_S^\sharp(X,Y) \subset Map^\sharp(X,Y)

to be the subcomplexes spanned by the cells that respect that map to the base $S$.

###### Observation

Let $Y\to S$ be a Cartesian fibration of simplicial sets, and ${X}^{♮}$ as above the marked simplicial set with precisely the Cartesian morphisms marked.

Then

• ${\mathrm{Map}}_{S}^{♭}\left(X,{Y}^{♮}\right)$ is an quasi-category;

• ${\mathrm{Map}}_{S}^{♯}\left(X,{Y}^{♮}\right)$ is its core, the maximal Kan complex inside it.

This is HTT, remark 3.1.3.1.

###### Proof

The $n$-cells of ${\mathrm{Map}}_{S}^{♭}\left(X,{Y}^{♮}\right)$ are morphisms $X×\Delta \left[n{\right]}^{♭}\to {Y}^{♮}$ over $S$. This means that for fixed $x\in {X}_{0}$, $\Delta \left[n\right]$ maps into a fiber of $Y\to S$. But fibers of Cartesian fibrations are fibers of inner fibrations, hence are quasi-categories.

Similarly, the $n$-cells of ${\mathrm{Map}}_{S}^{♯}\left(X,{Y}^{♮}\right)$ are morphisms $X×\Delta \left[n{\right]}^{♯}\to {Y}^{♮}$ over $S$. Again for fixed $x\in {X}_{0}$, $\Delta \left[n\right]$ maps into a fiber of $Y\to S$, 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.

###### Proposition

We have a sequence of adjoint functors

$\left(-{\right)}^{♭}⊣\left(-{\right)}^{♭}⊣\left(-{\right)}^{♮}⊣\left(-{\right)}^{♮}:\begin{array}{ccc}& \stackrel{\left(-{\right)}^{♭}}{\to }& \\ & \stackrel{\left(-{\right)}^{♭}}{←}& \\ \mathrm{sSet}& \stackrel{\left(-{\right)}^{♮}}{\to }& {\mathrm{sSet}}^{+}\\ & \stackrel{\left(-{\right)}^{♮}}{←}& \end{array}$(-)^{\flat} \dashv (-)^{\flat} \dashv (-)^{\natural} \dashv (-)^{\natural} : \array{ & \stackrel{(-)^{\flat}}{\to} & \\ & \stackrel{(-)^{\flat}}{\leftarrow} & \\ sSet & \stackrel{(-)^{\natural}}{\to} & sSet^+ \\ & \stackrel{(-)^{\natural}}{\leftarrow} & }

## Model structure on marked simplicial sets

### Cartesian weak equivalences

Observe that weak equivalences in the model structure for quasi-categories may be characterized as follows.

###### Lemma

A morphism $f:C\to D$ between simplicial sets that are quasi-categories is a weak equivalence in the model structure for quasi-categories precisely if the following equivalent coditions hold:

• For every simplicial set $K$, the morphism $\mathrm{sSet}\left(K,f\right):\mathrm{sSet}\left(K,D\right)\to \mathrm{sSet}\left(K,C\right)$ is a weak equivalence in the model structure for quasi-categories.

• For every simplicial set $K$, the morphism $\mathrm{Core}\left(\mathrm{sSet}\left(K,f\right)\right):\mathrm{Core}\left(\mathrm{sSet}\left(K,D\right)\right)\to \mathrm{Core}\left(\mathrm{sSet}\left(K,C\right)\right)$ on the cores, the maximal Kan complexes inside, is a weak equivalence in the standard model structure on simplicial sets, hence a homotopy equivalence.

###### Proof

This is HTT, lemma 3.1.3.2.

This may be taken as motivation for the following definition.

###### Definition/Proposition

For every Cartesian fibration $Z\to S$, we have that

${\mathrm{Map}}^{♭}\left(X,{Z}^{♮}\right)$Map^\flat(X, Z^{\natural})

is a quasi-category and

${\mathrm{Map}}^{♯}\left(X,{Z}^{♮}\right)=\mathrm{Core}\left({\mathrm{Map}}^{♭}\left(X,{Z}^{♮}\right)\right)$Map^\sharp(X,Z^\natural) = Core(Map^\flat(X,Z^\natural))

is the maximal Kan complex inside it.

A morphism $p:X\to Y$ in ${\mathrm{sSet}}^{+}/S$ is a Cartesian equivalence if for every Cartesian fibration $Z$ we have

• The induced morphism ${\mathrm{Map}}_{S}^{♭}\left(Y,{Z}^{♮}\right)\to {\mathrm{Map}}_{S}^{♭}\left(X,{Z}^{♮}\right)$ is an equivalence of quasi-categories;

Or equivalently:

• The induced morphism ${\mathrm{Map}}_{S}^{♯}\left(Y,{Z}^{♮}\right)\to {\mathrm{Map}}_{S}^{♯}\left(X,{Z}^{♮}\right)$ is a weak equivalence of Kan complexes.

This is HTT, prop. 3.1.3.3 with HTT, remark 3.1.3.1.

###### Proposition

Let

$\begin{array}{ccccc}X& & \stackrel{p}{\to }& & Y\\ & ↘& & ↙\\ & & S\end{array}$\array{ X &&\stackrel{p}{\to}&& Y \\ & \searrow && \swarrow \\ && S }

be a morphism in $\mathrm{sSet}/S$ such that both vertical maps to $S$ are Cartesian fibrations. Then the following are equivalent:

• $p$ is a homotopy equivalence.

• The induced morphism ${X}^{♮}\to {Y}^{♮}$ in ${\mathrm{sSet}}^{+}/S$ is a Cartesian equivalence.

• The induced morphism on each fiber ${X}_{s}\to {Y}_{p\left(s\right)}$ is a weak equivalence in the model structure for quasi-categories.

###### Proof

This is HTT, lemma 3.1.3.5.

### 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

(Cartesian 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

${\mathrm{sSet}}^{+}/S\left(X,Y\right):={\mathrm{Map}}_{S}^{♯}\left(X,Y\right)\phantom{\rule{thinmathspace}{0ex}}.$sSet^+/S(X,Y) := Map_S^\sharp(X,Y) \,.

A morphism $f:X\to X\prime$ in ${\mathrm{SSet}}^{+}/S$ of marked simplicial sets is

###### Proof

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

###### Remark

Using ${\mathrm{Map}}_{S}^{♭}\left(X,Y\right)$ for the mapping objects makes ${\mathrm{sSet}}^{+}/S$ a ${\mathrm{sSet}}_{\mathrm{Joyal}}$-enriched model category (i.e. enriched in the model structure for quasi-categories). This is HTT, remark 3.1.4.5.

Notice that trivially every object in this model structure is cofibrant. The following proposition shows that the above model structure indeed presents the $\left(\infty ,1\right)$-category $\mathrm{CartFib}\left(S\right)$ of Cartesian fibrations.

###### Proposition

An object $p:X\to S$ in ${\mathrm{sSet}}^{+}/S$ is fibrant with respect to the above model structure precisely if it is isomorphic to an object of the form ${Y}^{♮}$, for $Y\to S$ a Cartesian fibration in sSet.

###### Proof

This is HTT, prop. 3.1.4.1.

In particular, the fibrant objects of ${\mathrm{sSet}}^{+}\cong {\mathrm{sSet}}^{+}/*$ are precisely the quasicategories in which the marked edges are precisely the equivalences. Note that the Cartesian model structure on ${\mathrm{sSet}}^{+}/S$ is not the model structure on an over category induced on ${\mathrm{sSet}}^{+}/S$ from the Cartesian model structure on ${\mathrm{sSet}}^{+}$!

###### Definition/Proposition

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

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

### Marked anodyne morphisms

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 ${\mathrm{sSet}}^{+}$ 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.

###### Definition (HTT, Def 3.1.1.1)

The collection of marked anodyne morphisms in ${\mathrm{SSet}}^{+}/S$ is the class of morphisms ${\mathrm{An}}^{+}=\mathrm{LLP}\left(\mathrm{RLP}\left({\mathrm{An}}_{0}^{+}\right)\right)$ where the generating set ${\mathrm{An}}_{0}^{+}$ consists of

• for $0 the minimally marked horn inclusions

$\left(\Lambda \left[n{\right]}_{i}{\right)}^{♭}\to \Delta \left[n{\right]}^{♭}$(\Lambda[n]_i)^\flat \to \Delta[n]^\flat
• for $i=n$ the horn inclusion with the last edge markeed:

$\left(\Lambda \left[n{\right]}_{i},ℰ\cap \left(\Lambda \left[n{\right]}_{i}{\right)}_{1}\right)\to \left(\Delta \left[n\right],ℰ\right)\phantom{\rule{thinmathspace}{0ex}},$(\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 $\Delta \left[n\right]$ together with the edge ${\Delta }^{\left\{n-1,n\right\}}\to \Delta \left[n\right]$.

• the inclusion

$\left(\Lambda \left[2{\right]}_{1}{\right)}^{♮}\coprod _{\left(\Lambda \left[2{\right]}_{1}{\right)}^{♭}}\left(\Delta \left[2\right]{\right)}^{♭}\to \left(\Delta \left[2\right]{\right)}^{♮}\phantom{\rule{thinmathspace}{0ex}}.$(\Lambda[2]_1)^\natural \coprod_{(\Lambda[2]_1)^\flat} (\Delta[2])^\flat \to (\Delta[2])^\natural \,.
• for every Kan complex $K$ the morphism ${K}^{♭}\to {K}^{♯}$.

The crucial property of marked anodyne morphisms is the following characterization of morphisms that have the right lifting property with respect to them.

###### Proposition

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 $p$-Cartesian morphism and $p\left(e\right)$ is marked in $S$

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

###### Proof

This is HTT, prop. 3.1.1.6

###### Remark

Thus, if $\left(X,{E}_{X}\right)\to \left(S,{E}_{S}\right)$ is a morphism in ${\mathrm{sSet}}^{+}$ with RLP against marked anodyne morphisms, then its underlying morphism $X\to S$ in $\mathrm{sSet}$ is almost a Cartesian fibration: it may fail to be such only due to missing markings in ${E}_{S}$.

However, if all morphisms in $S$ are marked, then $\left(X,{E}_{X}\right)\to {S}^{♯}$ has the RLP against marked anodyne morphisms precisely when the underlying morphism $X\to S$ is a Cartesian fibration and exactly the Cartesian morphisms are marked in $X$, $\left(X,{E}_{X}\right)={X}^{♮}$ — in other words, precisely if it is a fibrant object in the model structure on ${\mathrm{sSet}}^{+}/S$.

The following stability property of marked anodyne morphisms is important in applications. Recall that a cofibration in ${\mathrm{sSet}}^{+}$ is a morphism whose underlying morphism in sSet is a monomorphism.

###### Proposition

(stability under smash product with cofibrations)

Marked anodyne morphisms are stable under “smash product” with cofibrations:

for $f:X\to X\prime$ marked anodyne, and $g:Y\to Y\prime$ a cofibration, the induced morphism

$\left(X×Y\prime \right)\coprod _{X×Y}\left(X\prime ×Y\right)\to X\prime ×Y\prime$(X \times Y') \coprod_{X \times Y} (X' \times Y) \to X' \times Y'

out of the pushout in ${\mathrm{sSet}}^{+}$ is marked anodyne.

###### Proof

This is HTT, prop. 3.1.2.3.

### As a model for the $\left(\infty ,1\right)$-category of $\left(\infty ,1\right)$-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 $\left({\mathrm{sSet}}_{\mathrm{Joyal}}{\right)}^{\circ }$ 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.

## References

Marked simplicial sets are introduced in section 3.1 of

The model structure on marked simplicial oversets is described in section 3.1.3

Revised on January 13, 2013 00:10:28 by Stephan Alexander Spahn (192.87.226.73)