nLab right/left Kan fibration

Idea

For ordinary categories there is the notion of

1. Grothendieck fibration between two categories,

2. and the special case of a fibration fibered in groupoids.

The analog of this for quasi-categories are

1. Cartesian fibrations

2. the special case of left/right (Kan-) fibrations of quasi-categories

respectively.

Definition

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 $\Lambda[n]_k \to \Delta[n]$ except possibly the right outer ones: $0 \leq k \lt n$.

It is a right fibration or right Kan fibration if its extends against all horns except possibly the left outer ones: $0 \lt k \leq n$.

So $X \to S$ is a left fibration precisely if for all commuting squares

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

for $n \in \mathbb{N}$ and $0 \leq k \lt n$, a diagonal lift exists as indicated.

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

Motivation: ordinary fibrations in groupoids are right Kan fibrations

Ordinary categories fibered in groupoids have a simple characterization in terms of their nerves. Let $N : Cat \to sSet$ be the nerve functor and for $p : E \to B$ a morphism in Cat (a functor), let $N(p) : N(E) \to N(B)$ be the corresponding morphism in sSet.

Then

Proposition

A functor $p \colon E \to B$ is a fibration in groupoids precisely if its nerve $N(p) \colon N(E) \to N(B)$ is a right Kan fibration of simplicial sets.

To see this, first notice the following facts:

Lemma 1

For $C$ a category, the nerve $N(C)$ is 2-coskeletal. In particular all $n$-spheres for $n \geq 3$ have unique fillers

$$\array{ \partial \Delta[n] &\stackrel{\forall}{\to}& N(C) \\ \downarrow & \nearrow_{\mathrlap{\exists !}} \\ \Delta[n] } \;\;\;\;\; (n \geq 3)$$

and (implied by that) all $n$-horns for $n \gt 3$ have fillers

$$\array{ \partial \Lambda[n] &\stackrel{\forall}{\to}& N(C) \\ \downarrow & \nearrow_{\mathrlap{\exists }} \\ \Delta[n] } \;\;\;\;\; (n \gt 3) \,.$$

This is discussed at nerve.

Lemma 2

If $p : E \to B$ is an ordinary functor, then $N(f) : N(E) \to N(B)$ is an inner fibration, meaning that its has the right lifting property with respect to all inner horn inclusions $\Lambda[n]_i \hookrightarrow \Delta[n]$ for $0 \lt i \lt n$.

This is discussed at inner fibration.

Proof of the proposition

From the above lemmas it follows that $N(p) : N(E) \to N(B)$ is a right fibration already precisely if it has the right lifting property with respect only to the three horn inclusions

$$\{ \Lambda[n]_n \hookrightarrow \Delta[n] | n = 1,2,3\} \,.$$

So we check explicitly what these three conditions amount to

• $n=1$ – The existence of all fillers

  $$\array{ \Lambda[1]_1 = \Delta^{\{1\}} &\stackrel{e}{\to}& N(E) \\ \downarrow & {}^{{\hat f}}\nearrow & \downarrow^{\mathrlap{N(p)}} \\ \Delta^{\{0 \to 1\}} &\stackrel{f}{\to}& N(B) }$$

  means that for all objects $e \in E$ and morphism $f : b \to p(e)$ in $B$, there exists a morphism $\hat f : \hat b \to e$ in $E$ such that $p(\hat f) = f$.

• $n=2$ – The existence of fillers

  $$\array{ \Lambda[2]_2 &\stackrel{e}{\to}& N(E) \\ \downarrow & {}^{{\hat f}}\nearrow & \downarrow^{\mathrlap{N(p)}} \\ \Delta[2] &\stackrel{f}{\to}& N(B) }$$

  means that for all diagrams

  $$\array{ && e_1 \\ &&& \searrow^{\mathrlap{\epsilon_{12}}} \\ e_0 &&\stackrel{\epsilon_{02}}{\to}&& e_2 }$$

  in $E$ and commuting triangles

  $$\array{ && p(e_1) \\ & {}^{\mathllap{f}}\nearrow&& \searrow^{\mathrlap{p(\epsilon_{12}})} \\ p(e_0) &&\stackrel{p(\epsilon_{02})}{\to}&& p(e_2) }$$

  in $B$, there is a commuting triangle

  $$\array{ && e_1 \\ &{}^{\mathllap{\hat f}}\nearrow&& \searrow^{\mathrlap{\epsilon_{12}}} \\ e_0 &&\stackrel{\epsilon_{02}}{\to}&& e_2 }$$

  in $E$, such that $p(\hat f ) = f$.

• $n=3$ – …

  Consider first the case of degenrate 3-simplices on $N(B)$, on 2-simplices as above.

  Suppose in the above situation two lifts $(\hat f)_1$ and $(\hat f)_2$ are found. Together these yield a $\Lambda[3]_3$-horn in $N(E)$. The filler condition says this can be filled, which implies that $(\hat f)_1 = (\hat f)_2$.

  So the $n=3$-condition implies that the lift whose existence is guaranteed by the $n=2$-condition is unique.

  By similar reasoning one sees that this is all the $n=3$-condition yields.

In total, these three lifting conditions are precisely those for a Grothendieck fibration in groupoids.

Properties

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

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

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.

By the pasting law for pullbacks this implies

Corollary

Let $p : X \to S$ be a left fibration of simplicial sets, and $f : Y \to X$ be a morphism of simplicial sets. Then $f$ is a left fibration iff $p f$ is a left fibration

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

Over a Kan complex $T$, left fibrations $S \to T$ are automatically Kan fibrations.

Proof

This appears as HTT, prop. 2.1.3.3.

As an important special case:

Corollary

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

Proof

This is originally due to Andre Joyal. Recalled at HTT, prop. 1.2.5.1.

Proposition

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

Proof

This follows on general grounds, since they are defined by a right lifting property (see, e.g. hereTheory#ClosurePropertiesOfInjectiveAndProjectiveMorphisms))

Proposition

The pullback (in SimplicialSets) of a left or right fibration is a homotopy pullback in the Joyal model structure for quasi-categories

(Cisinski 2019, Cor. 5.3.6)

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 propjection $C_{p/} \to C$ is a left fibration.

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

(Left/)Right anodyne morphisms

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 $LAn_0$:

$LAn_0 =$

1. the collection of all left horn inclusions

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

1. the collection of all inclusions of the form

   $$(\Delta[m] \times \{0\}) \coprod_{\partial \Delta[m] \times \{0\}} (\partial \Delta[m] \times \Delta[1]) \hookrightarrow \Delta[m] \times \Delta[1]$$

2. the collection of all inclusions of the form

   $$(S' \times \{0\}) \coprod_{S \times \{0\}} (S \times \Delta[1]) \hookrightarrow S' \times \Delta[1]$$

   for all inclusions of simplicial sets $S \hookrightarrow S'$.

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

Proof

…

Corollary

For $i : A \to A'$ left-anodyne and $j : B \to B'$ a cofibration in the model structure for quasi-categories, the canonical morphism

$$(A \times B') \coprod_{A \times B} (A' \times B) \to A' \times B'$$

is left-anodyne.

This appears as HTT, cor. 2.1.2.7.

Corollary

For $p : X \to S$ a left fibration and $i : A \to B$ a cofibration of simplicial sets, the canonical morphism

$$q : X^B \to X^A \times_{S^A} S^B$$

is a left fibration. If $i$ is furthermore left anodyne, then it is an acyclic Kan fibration.

This appears as HTT, cor. 2.1.2.9.

Proposition

For $f : A_0 \to A$ and $g : B_0 \to B$ two inclusions of simplicial sets with $f$ left anodyne, we have that the canonical morphism

$$(A_0 \star B ) \coprod_{A_0 \star B_0} (A \star B_0) \to A \star B$$

into the join of simplicial sets is left anodyne.

This is due to Andre Joyal. It appears as HTT, lemma 2.1.4.2.

Proposition

(restriction of over-quasi-categories along left anodyne inclusions)

Let $p : B \to S$ be a morphism of simplicial sets and $i : A \to B$ a left anodyne morphism, then the restriction morphism of under quasi-categories

$$S_{/p} \to S_{/p|_A}$$

is an acyclic Kan fibration.

This is a special case of what appears as HTT, prop. 2.1.2.5, which is originally due to Andre Joyal.

Proposition

Let $p : X \to S$ be a morphism of simplicial sets with section $s : S \to X$. If there is a fiberwise simplicial homotopy $X \times \Delta[1] \to S$ from $s \circ p$ to $Id_X$ then $s$ is left anodyne.

This appears as HTT, prop. 2.1.2.11.

Related concepts

• Kan fibration, anodyne morphism

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

  • model structure for left fibrations

  • universal right fibration

• inner fibration

• Cartesian fibration

• right/left fibration of simplicial spaces

References

• Jacob Lurie, Section 2.1 of: Higher Topos Theory, Annals of Mathematics Studies 170, Princeton University Press 2009 (pup:8957, pdf)

• David Ayala, John Francis, Fibrations of $\infty$-Categories (arXiv:1702.02681)

• Denis-Charles Cisinski, Section 3.4 and 5.3 of: Higher Categories and Homotopical Algebra, Cambridge University Press 2019 (doi:10.1017/9781108588737, pdf)