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,

rspectively.

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: $0\le k<n$.

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

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

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

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

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

Write

$$\mathrm{RFib}(S)\subset \mathrm{sSet}/S$$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 $N:\mathrm{Cat}\to \mathrm{sSet}$ 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

The functor $p:E\to B$ is an fibration in groupoids precisely if the morphism $N(p):N(E)\to N(B)$ 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 $n\ge 3$ have unique fillers

$$\begin{array}{ccc}\partial \Delta [n]& \stackrel{\forall }{\to }& N(C)\\ \downarrow & {\nearrow }_{\exists !}\\ \Delta [n]\end{array}(n\ge 3)$$\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 $n>3$ have fillers

$$\begin{array}{ccc}\partial \Lambda [n]& \stackrel{\forall }{\to }& N(C)\\ \downarrow & {\nearrow }_{\exists }\\ \Delta [n]\end{array}(n>3).$$\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 $0<i<n$.

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

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

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

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

  $$\begin{array}{ccc}& & e_1\\ & & & {\searrow }^{{ϵ}_{12}}\\ e_0& & \stackrel{{ϵ}_{02}}{\to }& & e_2\end{array}$$\array{ && e_1 \\ &&& \searrow^{\mathrlap{\epsilon_{12}}} \\ e_0 &&\stackrel{\epsilon_{02}}{\to}&& e_2 }

  in $E$ and commuting triangles

  $$\begin{array}{ccc}& & p(e_1)\\ & {}^f\nearrow & & {\searrow }^{p({ϵ}_{12})}\\ p(e_0)& & \stackrel{p({ϵ}_{02})}{\to }& & p(e_2)\end{array}$$\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

  $$\begin{array}{ccc}& & e_1\\ & {}^{\hat{f}}\nearrow & & {\searrow }^{{ϵ}_{12}}\\ e_0& & \stackrel{{ϵ}_{02}}{\to }& & e_2\end{array}$$\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 $p^{\mathrm{op}}:C^{\mathrm{op}}\to D^{\mathrm{op}}$ is a right fibration.

Therefore it is sufficient to list properties of only one type of these fib rations, 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 $Y\stackrel{(\mathrm{Id},0)}{\to }Y\times I$ for $I=[0,1]$ the standard interval and every commuting diagram

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

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

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

$$X^{\Delta [1]}\to X^{\{0\}}{\times }_{S^{\{0\}}}{S}^{{\Delta }^1}$$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 $\mathrm{\infty }$-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 of this we have

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.

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

$$\begin{array}{ccc}{X}_c& \to & X\\ \downarrow & & \downarrow \\ \{c\}& \to & C\end{array}$$\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 $(\mathrm{\infty },1)$-category it is an $\mathrm{\infty }$-groupoid. Hence fixing only one fiber of the bifibration should yield a fibration in $\mathrm{\infty }$-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 $\mathrm{LAn}=\mathrm{LLP}(\mathrm{RLP}({\mathrm{LAn}}_0))$ for the following choices of ${\mathrm{LAn}}_0$:

${\mathrm{LAn}}_0=$

1. the collection of all left horn inclusions

$\{\Lambda [n{]}_i\to \Delta [n]\mid 0\le i<n\}$;

1. the collection of all inclusions of the form

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

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

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

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

Proof

…

Corollary

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

$$(A\times B\prime )\coprod _{A\times B}(A\prime \times B)\to A\prime \times B\prime$$(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

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

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

$$S_{/p}\to S_{/p{\mid }_A}$$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 ${\mathrm{Id}}_X$ 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