nLab fibration of quasi-categories

Idea

For ordinary categories there is the notion of

• Grothendieck fibration between two categories.

• and the special case of a category fibered in groupoids.

The analog of this for quasi-categories are

• Cartesian fibrations

• left fibrations/right fibrations of quasi-categories.

There are more types of fibrations between the simplicial sets underlying the quasi-category

• inner fibrations– these correspond to bundles of quasi-categories : an inner fibration $E \to \Delta[1]$ over the interval is the quasi-categorical analog of a cograph of a profunctor: it characterizes the fibers $C, D$ over the endpoints $0,1 \in \Delta[1]$ as quasi-categories. Notably having an inner fibration $C \to \Delta[0]$ over the point says precisely that $C$ is a quasi-category.

• categorical fibrations – these appear as the fibrations in the sense of model category theory in the Joyal model structure for quasi-categories $sSet_{Joyal}$ . But they have no particular intrinsic meaning in higher category theory.

• minimal Joyal fibrations

Definition

We list the different definitions in the order of their generality. The examples of each definition are also examples of the following definitions.

All morphisms in the following are morphisms of simplicial sets.

Trivial fibration

A trivial fibration (trivial Kan fibration) is a morphism that has the right lifting property with respect to the boundary inclusions $\partial \Delta[n] \hookrightarrow \Delta[n], n \geq 1$.

Kan fibration

• for the moment see Kan fibration

A morphism with left lifting property against all Kan fibrations is called anodyne.

(Left/)Right fibration

A morphism of simplicial sets $f : X \to S$ is a left fibration or left Kan fibration if it has the right lifting property with respect to all horn inclusions except the right outer ones. It is a right fibration or right Kan fibration if its extends against all horns except the left outer ones.

$$\array{ \Lambda[n]_{k \gt 0} &\to& X \\ \downarrow &{}^{\exists}\nearrow& \downarrow \\ \Delta[n] &\to& S } \,.$$

Morphisms with the left lifting property against all left/right fibrations are called left/right anodyne morphism maps.

Write

$$RFib(S) \subset sSet/S$$

for the full SSet-subcategory of the overcategory of SSet over $S$ on those morphisms that are right fibrations.

This is a Kan complex-enriched category and as such an incarnation of the (∞,1)-category of right fibrations. It is modeled by the model structure for right fibrations. For details on this see the discussion at (∞,1)-Grothendieck construction.

(co)Cartesian fibration

A Cartesian fibration is an inner fibration $p : X \to S$ such that

• for every edge $f : X \to Y$ of $S$

• and every lift $\tilde{y}$ of $y$ (that is, $p(\tilde{y})=y$),

there is a Cartesian edge $\tilde{f} : \tilde{x} \to \tilde{y}$ in $X$ lifting $f$.

(HTT, def 2.4.2.1)

see also

• model structure for Cartesian fibrations

• (∞,1)-Grothendieck construction

Categorical fibration

A categorical fibration is a fibration in the model structure for quasi-categories: morphism $f : X \to S$ with the right lifting property against all monomorphic categorical equivalences .

(HTT, p. 81).

Inner fibration

A morphism of simplicial sets $f : X \to S$ is an inner fibration or inner Kan fibration if its has the right lifting property with respect to all inner horn inclusions.

The morphisms with the left lifting property against all inner fibrations are called inner anodyne.

Minimal fibration

• minimal Kan fibration

• minimal Joyal fibration

Properties

Remark

By the small object argument we have that every morphism $f : X \to Y$ of simplicial sets may be factored as

$$f : X \to Z \to Y$$

with $X \to Z$ a left/right/inner anodyne cofibration and $Z \to Y$ accordingly a left/right/inner Kan fibration.

Trivial fibration

…

Kan fibration

…

(Left/)Right fibration

Remark

Under the operation of forming the opposite quasi-category, left fibrations turn into right fibrations, and vice versa: if $p : C \to D$ is a left fibration then $p^{op} : C^{op} \to D^{op}$ is a right fibration.

Therefore it is sufficient to list properties of only one type of these fibrations, that for the other follows.

Homotopy lifting property

In classical homotopy theory, a continuous map $p : E \to B$ of topological spaces is said to have the homotopy lifting property if it has the right lifting property with respect to all morphisms $Y \stackrel{(Id, 0)}{\to} Y \times I$ for $I = [0,1]$ the standard interval and every commuting diagram

$$\array{ Y &\to& E \\ \downarrow && \downarrow \\ Y \times I &\to& B }$$

there exists a lift $\sigma : Y \times I \to E$ making the two triangles

$$\array{ Y &\to& E \\ \downarrow &{}^\sigma\nearrow& \downarrow \\ Y \times I &\to& B }$$

commute. For $Y = *$ the point this can be rephrased as saying that the universal morphism $E^I \to B^I \times_B E$ induced by the commuting square commuting square

$$\array{ E^I &\to& E \\ \downarrow && \downarrow \\ B^I &\to& B }$$

is an epimorphism. If it is even an isomorphism then the lift $\sigma$ exists uniquely . This is the situation that the following proposition generalizes:

Proposition

A morphism $p : X \to S$ of simplicial sets is a left fibration precisely if the canonical morphism

$$X^{\Delta[1]} \to X^{\{0\}} \times_{S^{\{0\}}} S^{\Delta^1}$$

is a trivial Kan fibration.

Proof

This is a corollary of the characterization of left anodyne morphisms in Properties of left anodyne maps by Andre Joyal, recalled in HTT, corollary 2.1.2.10.

As fibrations in $\infty$-groupoids

The notion of right fibration of quasi-categories generalizes the notion of category fibered in groupoids. This follows from the following properties.

Proposition

For $C \to *$ a right (left) fibration over the point, $C$ is a Kan complex, i.e. an ∞-groupoid.

Proof

Due to Andre Joyal. Recalled at HTT, prop. 1.2.5.1.

Proposition

Right (left) fibrations are preserved by pullback in sSet.

Corollary

It follows that the fiber $X_c$ of every right fibration $X \to C$ over every point $c \in C$, i.e. the pullback

$$\array{ X_c &\to& X \\ \downarrow && \downarrow \\ \{c\} &\to& C }$$

is a Kan complex.

Proposition

For $C$ and $D$ quasi-categories that are ordinary categories (i.e. simplicial sets that are nerves of ordinary categories), a morphism $C \to D$ is a right fibration precisely if the correspunding ordinary functor exhibits $C$ as a category fibered in groupoids over $D$.

Proof

This is HTT, prop. 2.1.1.3.

A canonical class of examples of a fibered category is the codomain fibration. This is actually a bifibration. For an ordinary category, a bifiber of this is just a set. For an $(\infty,1)$-category it is an $\infty$-groupoid. Hence fixing only one fiber of the bifibration should yield a fibration in $\infty$-groupoids. This is asserted by the following statement.

Proposition

Let $p : K \to C$ be an arbitrary morphism to a quasi-category $C$ and let $C_{p/}$ be the corresponding under quasi-category. Then the canonical projection $C_{p/} \to C$ is a left fibration.

Proof

Due to Andre Joyal. Recalled as HTT, prop 2.1.2.2.

(Left/)Right anodyne moprphisms

Proposition

The collection of left anodyne morphisms (those with left lifting property against left fibrations) is equivalently $LAn = LLP(RLP(LAn_0))$ for the following choices of $LAn_0$:

$LAn_0 =$

• the collection of all left horn inclusions

  $\{ \Lambda[n]_{i} \to \Delta[n] | 0 \leq i \lt n \}$;

• blah-blah

• blah-blah

Proof

This is due to Andre Joyal, recalled as HTT, prop 2.1.2.6.

(co)Cartesian fibration

Categorical fibration

Inner fibrations

Proposition

A simplicial set $K$ is the nerve of an ordinary category $C$, $K \simeq_{iso} N(C)$ precisely if the terminal morphism $K \to \Delta[0]$ is an inner fibration with unique inner horn fillers, i.e. precisely if for all morphisms

$$\Lambda[n]_i \to K$$

with $n \in \mathbb{N}$ and $0 \lt i \lt n$ there is a unique morphism $\Delta[n] \to K$ making the diagram

$$\array{ \Lambda[n]_i &\to& K \\ \downarrow & \nearrow \\ \Delta[n] }$$

commute.

Proof

This is HTT, prop. 1.1.2.2

Corollary

It follows that under the nerve every functor $f : C \to D$ between ordinary categories is an inner fibration.

Proof

This is immediate, but let’s spell it out:

In any commutative diagram

$$\array{ \Lambda[n] &\to& N(C) \\ \downarrow && \downarrow^{\mathrlap{N(f)}} \\ \Delta[n] && \to& N(D) }$$

by the above the bottom morphism is already uniquely specified by the remaining diagram.

$$\array{ \Lambda[n] &\to& N(C) \\ \downarrow && \downarrow^{\mathrlap{N(f)}} \\ \Delta[n] && N(D) } \,.$$

By the above there exists a unique lift into $N(C)$

$$\array{ \Lambda[n] &\to& N(C) \\ \downarrow &\nearrow& \downarrow^{\mathrlap{N(f)}} \\ \Delta[n] && N(D) }$$

and by uniqueness of lifts into $N(D)$ this must also make the lower square commute

$$\array{ \Lambda[n] &\to& N(C) \\ \downarrow &\nearrow& \downarrow^{\mathrlap{N(f)}} \\ \Delta[n] &\to& N(D) } \,.$$

Related concepts

• fibered category

• $n$-fibration

• universal fibration of (∞,1)-categories

References

• Jacob Lurie, chapter 2 of: Higher Topos Theory

• David Ayala and John Francis, Fibrations of ∞-categories, (arxiv:1702.02681)

• Clark Barwick, Jay Shah, Fibrations in ∞-category theory (arXiv:1607.04343)

Discussion in homotopy type theory:

• Ulrik Buchholtz, Jonathan Weinberger, Synthetic fibered (∞,1)-category theory [arXiv:2105.01724]