nLab Cartesian fibration

Idea

Cartesian fibrations are one of the types of fibrations of quasi-categories.

A Cartesian fibration of quasi-categories – or more generally of simplicial sets – is a morphism that generalizes the notion of Grothendieck fibration from category theory to (∞,1)-category theory, specifically with (∞,1)-categories incarnated as quasi-categories:

It is an inner fibration that lifts also all right outer horn inclusions whose last edge is a cartesian morphism, and such that there is a sufficient supply of cartesian morphisms.

An inner fibration $p : C \to D$ may be thought of as a family of (infinity,1)-categories $(C_d)_{d \in D}$ which is functorial in $d$ only in the sense of correspondences. Then the condition of $p$ being a cartesian fibration ensures that the family is actually functorial. More precisely, if an (∞,1)-functor $p : C \to D$ is a Cartesian fibration, then it is possible to interpret its value over any morphism $f : d_1 \to d_2$ in $D$ as an (∞,1)-functor $p^{-1}(f) : p^{-1}(d_2) \to p^{-1}(d_1)$ between the fibers $p^{-1}(d_2)$ and $p^{-1}(d_1)$ over its source and target objects.

This way a Cartesian fibration $p : C \to D$ determines and is determined by an (∞,1)-functor $D^{op} \to (\infty,1)Cat$ into the (∞,1)-category of (∞,1)-categories. This is the content of the (∞,1)-Grothendieck construction.

Cartesian fibrations over $S$ are the fibrant objects in the model structure on marked simplicial over-sets over $S$.

Definition

Definition

A morphism $p : X \to Y$ in sSet is a Cartesian fibration (or Grothendieck fibration) if

• it is an inner fibration

• for every edge $f : x \to y$ of $Y$ and for every lift $\hat {y}$ of $y$ (that is: $p(\hat{y})=y$) there is a Cartesian edge $\hat{f} : \hat{x} \to \hat{y}$ in $X$ lifting $f$ (that is: such that $p(\hat f) = f$)

The morphism is a cocartesian fibration (or Cartesian opfibration, Grothendieck opfibration) if the opposite $p^{op} : X^{op} \to Y^{op}$ is a Cartesian fibration.

This is HTT, def. 2.4.2.1.

Definition

Given cartesian fibrations $p : X \to C$ and $q : Y \to C$, let $Map_C^{cart}(p,q) \subseteq Map_C(p,q)$ and $Fun_C^{cart}(p,q) \subseteq Fun_C(p,q)$ be the full subsimplicial sets spanned by the functors that preserve cartesian morphisms. We call such functors cartesian functors.

Dually, we make the analogous definition of cocartesian functor.

Remark

The $Map_C^{cart}$ are the mapping spaces in the (cartesian) model structure on marked simplicial sets over $C$.

Properties

General properties

Proposition

We have:

• Every isomorphism of simplicial sets is a Cartesian fibration.

• The composite of two Cartesian fibrations is again a Cartesian fibration.

Proof

This is HTT, prop. 2.4.2.3.

Proposition

Every Cartesian fibration is a fibration in the Joyal-model structure for quasi-categories.

Proof

This is HTT, prop. 3.3.1.7.

Behaviour under pullback

Since a Cartesian fibration is in particular an inner fibration and since inner fibrations are stable under pullback in sSet, it follows that for $p : E \to C$ a Cartesian fibration, the fiber $E_x$ over every point $x \in C_0$ is a quasi-category

$$\array{ E_x &\to& E \\ \downarrow && \downarrow \\ \{x\} &\to& C } \,.$$

The difference between inner fibrations and Cartesian fibrations is that only for Cartesian fibrations is it generally guaranteed that these fibers over the points are functorially related over the morphisms in $C$. This is the content of the (∞,1)-Grothendieck construction.

But moreover:

Proposition

The pullback of a Cartesian fibration in sSet is again a Cartesian fibration.

Proof

This is HTT, prop. 2.4.2.3.

We know from the discussion at inner fibration that the pullback is an inner fibration. It remains to check if it has enough Cartesian morphisms. By HTT, prop 2.4.1.3 we have that in a pullback diagram

$$\array{ E' &\stackrel{q}{\to}& E \\ {}^{\mathllap{p'}}\downarrow && \downarrow^{\mathrlap{p}} \\ C' &\stackrel{k}{\to}& C }$$

a morphism $f \in E'$ is $p'$-Cartesian if $q(f)$ is $p$-Cartesian. Since the morphisms of $E'$ are pairs of morphisms $(\gamma, \hat f) \in C'_1 \times E_1$ and since by assumption $p$ is a Cartesian fibration, there is for $\gamma \in C'_1$ and $y \in E'_0$ such that $p'(y)$ is the target of the morphism $\gamma$ a Cartesian lift $\hat f \in E$ of $k(\gamma)$ such that $q(y)$ is the target of $\hat f$. Hence a Cartesian lift $(\gamma, \hat f)$ of $\gamma$ in $E'$ having $y$ as target.

We can test locally if a morphism is a Cartesian fibration:

Proposition

An inner fibration $p : X \to Y$ is Cartesian precisely if for each $n \leq 2$ and for every $n$-simplex $\sigma : \Delta[n] \to Y$ the sSet-pullback $\sigma^* p : X \times_Y \Delta[n] \to \Delta[n]$ in

$$\array{ X \times_Y \Delta[n] &\to& X \\ \downarrow && \downarrow^{\mathrlap{p}} \\ \Delta[n] &\stackrel{\sigma}{\to}& Y }$$

is a Cartesian fibration.

Proof

This is HTT, cor. 2.4.2.10.

Proposition

The pullback in sSet of a weak equivalence in the Joyal-model structure for quasi-categories along a Cartesian fibration is again a Joyal-weak equivalence

Proof

This is HTT, prop 3.3.1.3

Lemma

Equivalences in $sSet_{Joyal}$ are stable under pullback along Cartesian fibrations:

if

$$\array{ X \times_S T &\to & X \\ \downarrow && \downarrow \\ T &\stackrel{\simeq}{\to}& S }$$

is a pullback square in sSet with $T \to S$ a weak equivalence in $sSet_{Joyal}$ and $X \to S$ a Cartesian fibration, then $X \times_S T \to X$ is also a weak equivalence.

Proof

This is HTT, prop. 3.3.1.3.

The following proposition asserts that the ordinary pullback (in sSet) of Cartesian fibrations already models the correct homotopy pullback.

Proposition

Let

$$\array{ X &\to& X' \\ \downarrow && \downarrow \\ S &\to & S' }$$

be an pullback diagram in sSet of quasi-categories, where $X' \to S'$ is a Cartesian fibration. Then this is already a homotopy pullback diagram with respect to the model structure for quasi-categories.

Proof

This is HTT, prop 3.3.1.4. We factor the bottom morphism as $S \stackrel{\simeq}{\to} T \to S'$ into a weak equivalence and a fibration in $sSet_{Joyal}$. Then the right square in

$$\array{ X &\to& T \times_{S'} X'& \to& X' \\ \downarrow && \downarrow && \downarrow \\ S &\to & T &\to& S' }$$

is the ordinary pullback over a fibrant replacement of the original diagram hence is a homotopy pullback. The claim follows thus if $X \to T \times_{S'} X'$ is a weak equivalence, which it is by one of the above lemmas.

Straightening and unstraightening

There is a Quillen equivalence between the model category of cartesian fibrations and the model category of presheaves valued in quasicategories. See the article straightening functor for more information.

Relation to other kinds of fibrations

The notion of right fibration is a special case of that of Cartesian fibration:

Proposition

The following are equivalent:

• $p : X \to Y$ is a Cartesian fibration and every edge in $X$ is $p$-Cartesian

• $p$ is a right fibration;

• $p$ is a Cartesian fibration and every fiber is a Kan complex.

Proof

This is HTT, prop. 2.4.2.4.

There are also the “categorical fibrations”, the fibrations in the Joyal model structure for quasi-categories on $sSet$. These turn out not to have much of an intrinsic category theoretic meaning. By the following proposition one can understand the notion of Cartesian fibration as a suitable refinement of the notion of categorical fibration

Proposition

Every Cartesian fibration $p : X \to Y$ in sSet is a fibration with respect to the Joyal model structure for quasi-categories on sSet.

Proof

This is HTT, prop. 3.3.1.7.

However, when the base is an $\infty$-groupoid, then the difference between Cartesian fibrations and categorical fibrations disappears:

Proposition (over $\infty$-groupoids)

Let $p : X \to Y$ be a morphism of simplicial sets where $Y$ is a Kan complex. Then the following are equivalent:

1. $p$ is a Cartesian fibration

2. $p$ is a coCartesian fibration

3. $p$ is a categorical fibration

Proof

This is HTT, prop. 3.3.1.8.

There is however another model category structure, which does model Cartesian fibrations.

Proposition

Let $sSet^+/S$ be the overcategory of the category of marked simplicial sets over $S$, equipped with the model structure on marked simplicial over-sets.

An object $X \to S$ is fibrant in that model category precisely if

• the underlying morphism of simplicial sets $X \to S$ is a Cartesian fibration;

• the marked edges in $X$ are precisely the Cartesian morphisms.

Proof

This is HTT, prop. 3.1.4.1.

Adjoints to the inclusion into (∞,1)Cat/C

The inclusion of the cartesian fibrations and cartesian functors in the category $(\infty,1)Cat_{/C}^{cart} \subseteq (\infty,1)Cat_{/C}$ has both a left and a right adjoint.

The left adjoint is given by the construction of “free fibrations”

Proposition

For maps $p : X \to C$ and cartesian fibrations $q : Y \to C$, there are a natural equivalences

$$Fun_C(p, q) \simeq Fun_C^{cart}((C \downarrow p), q)$$

Dually, for maps $p : X \to C$ and cocartesian fibrations $q : Y \to C$, there are natural equivalences

$$Fun_C(p, q) \simeq Fun_C^{cocart}((p \downarrow C), q)$$

Proof

The cartesian case for mapping spaces is theorem 4.11 of Gepner-Haugseng-Nikolaus. For the cocartesian case,

$$Fun_C(p,q) \simeq Fun_{C^{op}}(p^{op}, q^{op})^{op} \simeq \Fun_{C^{op}}^{cart}((C^{op} \downarrow p^{op}), q^{op})^{op} \simeq \Fun_C^{cocart}((p \downarrow C), q)$$

On the other side,

Proposition

The inclusions $(\infty,1)Cat_{/C}^{cart} \subseteq (\infty,1)Cat_{/C}$ and $(\infty,1)Cat_{/C}^{cocart} \subseteq (\infty,1)Cat_{/C}$ have right adjoints.

Proof

We handle the cartesian case. The functor tensor product $- \otimes_C C_{/\bullet}$ is the left adjoint part of an adjunction between $Fun(C^{op}, (\infty,1)Cat)$ and $(\infty,1)Cat$. Since it sends $1 \mapsto C$, we also get an adjunction between $Fun(C^{op}, (\infty,1)Cat)$ and $(\infty,1)Cat_{/C}$

By the description of the (∞,1)-Grothendieck construction as a lax (∞,1)-colimit, the left adjoint part is an equivalence $Fun(C^{op}, (\infty,1)Cat) \to (\infty,1)Cat_{/C}^{cart}$

Examples

Projections from comma categories

Proposition

Given functors $F : A \to C$ and $G : B \to C$ between quasi-categories, the projection $(F \downarrow G) \to A$ is a Cartesian fibration and the projection $(F \downarrow G) \to B$ is a coCartesian fibration.

In particular, if $C$ is a quasi-category, then $eval_0 : C^{\Delta^1} \to C$ is a Cartesian fibration and $eval_1 : C^{\Delta^1} \to C$ is a coCartesian fibration.

Proof

Consider the diagram of pullback squares

$$\array{ (F \downarrow G) &\to& (id_C \downarrow G) &\to& B \\ \downarrow && \downarrow && \downarrow G\!\! \\ (F \downarrow id_C) &\to& C^{\Delta^1} &\xrightarrow{1}& C \\ \downarrow && \downarrow 0\!\! \\ A &\xrightarrow{F}& C }$$

HTT, prop. 2.4.7.12 states the projection $(id_C \downarrow G) \to C$ is a Cartesian fibration. Then $(F \downarrow G) \to A$ is as well, since fibrations are preserved by pullback. Dually for the other projection.

The fiber of $eval_1 : C^{\Delta^1} \to C$ at an object $X \in C$ is the slice category $C_{/X}$. So, by the (infinity,1)-Grothendieck construction, this fibration corresponds to the slice functor $C \to (\infty,1)Cat : X \mapsto C_{/X}$, where a morphism $X \to Y$ induces the composition map $C_{/X} \to C_{/Y}$.

Cartesian fibrations over one morphism and Adjunctions

By the (infinity,1)-Grothendieck construction a Cartesian fibration $K \to \Delta[1]$ corresponds to a morphism $\Delta[1]^{op} \to (\infty,1)Cat$, hence to an (infinity,1)-functor $f : C \to D$.

Obtaining this through the straightening functor above is tedious, but there is a more immediate way to characterize $f$:

Definition

For $p : K \to \Delta[1]$ a Cartesian fibration with specified equivalences

$$h_0 : C \stackrel{\simeq}{\to} p^{-1}(0)$$

and

$$h_1 : D \stackrel{\simeq}{\to} p^{-1}(1)$$

an (infinity,1)-functor $f : D \to C$ is associated to $p$ if there exists a commuting diagram

$$\array{ D \times \Delta[1] &&\stackrel{s}{\to}&& K \\ & \searrow && \swarrow \\ && \Delta[1] }$$

such that

• $s|_{D \times \{1\}} = h_1$;

• $s|_{D \times \{0\}} = h_0 \circ f$

• for every object $x$ of $D$ we have that $s|_{\{x\} \times \Delta[1]}$ is a p-Cartesian morphism of $K$.

This is HTT, def. 5.2.1.1.

If $p : K \to \Delta[1]$ is both a Cartesian fibration as well as a coCartesian fibration, then it determines $(\infty,1)$-functors in both directions

$$C \stackrel{\leftarrow}{\to} D \,.$$

Such a pair is a pair of adjoint (infinity,1)-functors.

Cartesian fibrations over simplices

… for the moment see HTT, section 3.2.2 …

The universal Cartesian fibration

for the moment see

• universal fibration of (∞,1)-categories.

Related concepts

• Kan fibration, anodyne morphism

• right/left Kan fibration, right/left anodyne map

• inner fibration

• Cartesian fibration, Cartesian fibration of dendroidal sets

• coCartesian fibration of (∞,1)-operads

• straightening functor

References

• Jacob Lurie, section 2.4.2 in Higher Topos Theory

• Aaron Mazel-Gee, A user’s guide to co/cartesian fibrations (arXiv:1510.02402)

• David Gepner, Rune Haugseng, Thomas Nikolaus, Lax colimits and free fibrations in $\infty$-categories (arXiv:1501.02161)

Discussion internal to any (∞,1)-topos:

• Louis Martini, Cocartesian fibrations and straightening internal to an ∞-topos [arXiv:2204.00295]