# nLab model structure for Cartesian fibrations

model category

## Model structures

for ∞-groupoids

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

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

(∞,1)-category theory

# Contents

## Idea

The model category structure on the category $SSet^+/S$ 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 $SSet^+/S$ 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 $sSet_{Quillen}$-enriched model category (i.e. enriched over the ordinary model structure on simplicial sets that models ∞-groupoids).

The $(\infty,1)$-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.

## 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 $(S,E)$ 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 $(S,E) \to (S',E')$ of marked simplicial sets is a morphism $f : S \to S'$ of simplicial sets that carries marked edges to marked edges in that $f(E) \subset E'$.

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

• for $S$ a simplicial set let

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

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

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

• $X^\natural$ or $X^{cart}$ 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 $sSet^+$ in a more abstract way.

###### Definition

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

$\array{ [0] &\stackrel{\to}{\to}& [1] \\ {}^{\mathllap{=}}\downarrow &\swarrow& \downarrow^{\mathrlap{p}} \\ [0] &\leftarrow& [1^+] } \,.$

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

###### Observation

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

$sSet^+ \simeq SepPSh(\Delta^+) \,.$
###### Proof

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

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

is a monomorphism, hence if $X_{1^+}$ is a subset of $X_1$. By functoriality this subset contains all the degenerate 1-cells

$\array{ X_0 \\ \downarrow & \searrow^{\mathrlap{\sigma}} \\ X_{1^+} &\hookrightarrow& X_1 } \,.$

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 $sSet^+$ is a genuine quasi-topos:

###### Observation

$sSet^+$ is not a topos.

###### Proof

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

### Cartesian closure

###### Lemma

The category $sSet^+$ is a cartesian closed category.

###### Proof

This is an immediate consequence of the above observation that $sSet^+$ 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 $PSh(\Delta^+)$ is cartesian closed. It remains to check that if $X,Y \in PSh(\Delta^+)$ 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

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

and the morphism $(Y^X)_{1^+} \to (Y^X)_1$ sends $X \times [1^+] \stackrel{\eta}{\to} Y$ to

$X \times [1] \stackrel{(Id,p)}{\to} X \times [1^+] \stackrel{\eta}{\to} Y \,.$

Now, by construction, every non-identity morphism $U \to [1^+]$ in $\Delta^+$ factors through $U \to [1]$, which implies that if the components of $p^* \eta_1$ and $p^* \eta_2$ coincide on $U \neq [1^+]$, 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 $U = [1^+]$ are already fixed, due to the inclusion $X_{1^+} \times [1^+]_{1^+} \hookrightarrow X_{1} \times [1^+]_{1}$. Hence $p^*$ is injective, and so $Y^X$ formed in $PSh(\Delta^+)$ is itself a marked simplicial set.

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

• $Map^\flat(X,Y)$ be the simplicial set underlying the cartesian internal hom $Y^X \in sSet^+$

• $Map^\sharp(X,Y)$ the simplicial set consisting of all simplices $\sigma \in Map^\flat(X,Y)$ 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

$Hom_{sSet}(K, Map^\flat(X,Y)) \simeq Hom_{sSet^+}(K^\flat, Y^X) \simeq Hom_{sSet^+}(K^\flat \times X, Y)$

and

$Hom_{sSet}(K, Map^\sharp(X,Y)) \simeq Hom_{sSet^+}(K^\sharp, Y^X) \simeq Hom_{sSet^+}(K^\sharp \times X, Y)$

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

$Map^\flat(X,Y)_n = Hom_{sSet^+}(X \times \Delta[n]^\flat, Y)$

and

$Map^\sharp(X,Y)_n = Hom_{sSet^+}(X \times \Delta[n]^\sharp, Y) \,.$

In words we have

• The $n$-simplices of the internal hom $Y^X$ are simplicial maps $X \times \Delta[n] \rightarrow Y$ such that when you restrict $X_1 \times \Delta[n]_1 \rightarrow Y_1$ to $E \times \Delta[n]_0$ (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 $X \times \Delta[1] \rightarrow Y$ such that the restriction of $X_1 \times \Delta[1]_1 \rightarrow Y_1$ to $E \times \Delta[1]_1$ factors though the marked edges of $Y$. In the presence of the previous condition, this says that when you apply the homotopy $X \times \Delta[1] \rightarrow Y$ to a marked edge of $X$ paired with the identity at $[1]$, the result should be marked.

###### Definition

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

$Map_S^\flat(X,Y) \subset Map^\flat(X,Y)$

and

$Map_S^\sharp(X,Y) \subset Map^\sharp(X,Y)$

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 $X^\natural$ as above the marked simplicial set with precisely the Cartesian morphisms marked.

Then

• $Map_S^\flat(X,Y^\natural)$ is an quasi-category;

• $Map_S^\sharp(X, Y^\natural)$ is its core, the maximal Kan complex inside it.

This is HTT, remark 3.1.3.1.

###### Proof

The $n$-cells of $Map_S^\flat(X,Y^\natural)$ are morphisms $X \times \Delta[n]^\flat \to Y^\natural$ over $S$. This means that for fixed $x \in X_0$, $\Delta[n]$ 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 $Map_S^\sharp(X,Y^\natural)$ are morphisms $X \times \Delta[n]^\sharp \to Y^\natural$ over $S$. Again for fixed $x \in X_0$, $\Delta[n]$ 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

$(-)^{\flat} \dashv (-)_{\flat} \dashv (-)^{\sharp} \dashv (-)_{\sharp} : \array{ & \stackrel{(-)^{\flat}}{\to} & \\ & \stackrel{(-)_{\flat}}{\leftarrow} & \\ sSet & \stackrel{(-)^{\sharp}}{\to} & sSet^+ \\ & \stackrel{(-)_{\sharp}}{\leftarrow} & }$

## 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 $sSet(K,f) : sSet(K,C) \to sSet(K,D)$ is a weak equivalence in the model structure for quasi-categories.

• For every simplicial set $K$, the morphism $Core(sSet(K,f)) : Core(sSet(K,C)) \to Core(sSet(K,D))$ 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

$Map_S^\flat(X, Z^{\natural})$

is a quasi-category and

$Map_S^\sharp(X,Z^\natural) = Core(Map_S^\flat(X,Z^\natural))$

is the maximal Kan complex inside it.

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

• The induced morphism $Map_S^\flat(Y,Z^{\natural}) \to Map_S^\flat(X,Z^{\natural})$ is an equivalence of quasi-categories;

Or equivalently:

• The induced morphism $Map_S^\sharp(Y,Z^{\natural}) \to Map_S^\sharp(X,Z^{\natural})$ is a weak equivalence of Kan complexes.

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

###### Proposition

Let

$\array{ X &&\stackrel{p}{\to}&& Y \\ & \searrow && \swarrow \\ && S }$

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

• $p$ is a homotopy equivalence.

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

• The induced morphism on each fiber $X_s \to Y_{p(s)}$ 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 $Set^+/S$ over $S \in SSet$ – also called the Cartesian model structure since it models Cartesian fibrations – is defined as follows.

###### Definition/Proposition

(Cartesian model structure on $sSet^+/S$)

The category $SSet^+/S$ 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

$sSet^+/S(X,Y) := Map_S^\sharp(X,Y) \,.$

A morphism $f : X \to X'$ in $SSet^+/S$ 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 $Map_S^\flat(X,Y)$ for the mapping objects makes $sSet^+/S$ a $sSet_{Joyal}$-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 $(\infty,1)$-category $CartFib(S)$ of Cartesian fibrations.

###### Proposition

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

###### Proof

This is HTT, prop. 3.1.4.1.

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

###### Definition/Proposition

(coCartesian model structure on $sSet^+/S$)

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 $sSet^+$ 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 $SSet^+/S$ is the class of morphisms $An^+ = LLP(RLP(An^+_0))$ where the generating set $An^+_0$ consists of

• for $0 \lt i \lt n$ the minimally marked horn inclusions

$(\Lambda[n]_i)^\flat \to \Delta[n]^\flat$
• for $i = n$ the horn inclusion with the last edge marked:

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

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

• the inclusion

$(\Lambda[2]_1)^\sharp \coprod_{(\Lambda[2]_1)^\flat} (\Delta[2])^\flat \to (\Delta[2])^\sharp \,.$
• for every Kan complex $K$ the morphism $K^\flat \to K^\sharp$.

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 $SSet^+$ has the right lifting property with respect to the class $An^+$ 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 $p(e)$ is marked in $S$

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

###### Proof

This is HTT, prop. 3.1.1.6

###### Remark

Thus, if $(X, E_X) \to (S,E_S)$ is a morphism in $sSet^+$ with RLP against marked anodyne morphisms, then its underlying morphism $X\to S$ in $sSet$ 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 $(X,E_X) \to S^\sharp$ 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 $X$, $(X,E_X) = X^\natural$ — in other words, precisely if it is a fibrant object in the model structure on $sSet^+/S$.

The following stability property of marked anodyne morphisms is important in applications. Recall that a cofibration in $sSet^+$ 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 $f : X \to X'$ marked anodyne, and $g : Y \to Y'$ a cofibration, the induced morphism

$(X \times Y') \coprod_{X \times Y} (X' \times Y) \to X' \times Y'$

out of the pushout in $sSet^+$ is marked anodyne.

###### Proof

This is HTT, prop. 3.1.2.3.

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

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

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

## 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

Last revised on July 27, 2016 at 20:57:42. See the history of this page for a list of all contributions to it.