# nLab right/left Kan fibration

### Context

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

(∞,1)-category theory

# Contents

## 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. 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 $\Lambda \left[n{\right]}_{k}\to \Delta \left[n\right]$ except possibly the right outer ones: $0\le k.

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

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

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

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

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

Write

$\mathrm{RFib}\left(S\right)\subset \mathrm{sSet}/S$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 $N\left(p\right):N\left(E\right)\to N\left(B\right)$ be the corresponding morphism in sSet.

Then

###### Proposition

The functor $p:E\to B$ is an fibration in groupoids precisely if the morphism $N\left(p\right):N\left(E\right)\to N\left(B\right)$ is a right Kan fibration of simplicial sets

To see this, first notice the following facts:

###### Lemma 1

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

$\begin{array}{ccc}\partial \Delta \left[n\right]& \stackrel{\forall }{\to }& N\left(C\right)\\ ↓& {↗}_{\exists !}\\ \Delta \left[n\right]\end{array}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\left(n\ge 3\right)$\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

$\begin{array}{ccc}\partial \Lambda \left[n\right]& \stackrel{\forall }{\to }& N\left(C\right)\\ ↓& {↗}_{\exists }\\ \Delta \left[n\right]\end{array}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\phantom{\rule{thickmathspace}{0ex}}\left(n>3\right)\phantom{\rule{thinmathspace}{0ex}}.$\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 $N\left(f\right):N\left(E\right)\to N\left(B\right)$ is an inner fibration, meaning that its has the right lifting property with respect to all inner horn inclusions $\Lambda \left[n{\right]}_{i}↪\Delta \left[n\right]$ for $0.

This is discussed at inner fibration.

###### Proof of the proposition

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

$\left\{\Lambda \left[n{\right]}_{n}↪\Delta \left[n\right]\mid n=1,2,3\right\}\phantom{\rule{thinmathspace}{0ex}}.$\{ \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

$\begin{array}{ccc}\Lambda \left[1{\right]}_{1}={\Delta }^{\left\{1\right\}}& \stackrel{e}{\to }& N\left(E\right)\\ ↓& {}^{\stackrel{^}{f}}↗& {↓}^{N\left(p\right)}\\ {\Delta }^{\left\{0\to 1\right\}}& \stackrel{f}{\to }& N\left(B\right)\end{array}$\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 $f:b\to p\left(e\right)$ in $B$, there exists a morphism $\stackrel{^}{f}:\stackrel{^}{b}\to e$ in $E$ such that $p\left(\stackrel{^}{f}\right)=f$.

• $n=2$ – The existence of fillers

$\begin{array}{ccc}\Lambda \left[2{\right]}_{2}& \stackrel{e}{\to }& N\left(E\right)\\ ↓& {}^{\stackrel{^}{f}}↗& {↓}^{N\left(p\right)}\\ \Delta \left[2\right]& \stackrel{f}{\to }& N\left(B\right)\end{array}$\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

$\begin{array}{ccc}& & {e}_{1}\\ & & & {↘}^{{ϵ}_{12}}\\ {e}_{0}& & \stackrel{{ϵ}_{02}}{\to }& & {e}_{2}\end{array}$\array{ && e_1 \\ &&& \searrow^{\mathrlap{\epsilon_{12}}} \\ e_0 &&\stackrel{\epsilon_{02}}{\to}&& e_2 }

in $E$ and commuting triangles

$\begin{array}{ccc}& & p\left({e}_{1}\right)\\ & {}^{f}↗& & {↘}^{p\left({ϵ}_{12}\right)}\\ p\left({e}_{0}\right)& & \stackrel{p\left({ϵ}_{02}\right)}{\to }& & p\left({e}_{2}\right)\end{array}$\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

$\begin{array}{ccc}& & {e}_{1}\\ & {}^{\stackrel{^}{f}}↗& & {↘}^{{ϵ}_{12}}\\ {e}_{0}& & \stackrel{{ϵ}_{02}}{\to }& & {e}_{2}\end{array}$\array{ && e_1 \\ &{}^{\mathllap{\hat f}}\nearrow&& \searrow^{\mathrlap{\epsilon_{12}}} \\ e_0 &&\stackrel{\epsilon_{02}}{\to}&& e_2 }

in $E$, such that $p\left(\stackrel{^}{f}\right)=f$.

• $n=3$ – …

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

Suppose in the above situation two lifts $\left(\stackrel{^}{f}{\right)}_{1}$ and $\left(\stackrel{^}{f}{\right)}_{2}$ are found. Together these yield a $\Lambda \left[3{\right]}_{3}$-horn in $N\left(E\right)$. The filler condition says this can be filled, which implies that $\left(\stackrel{^}{f}{\right)}_{1}=\left(\stackrel{^}{f}{\right)}_{2}$.

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 $Y\stackrel{\left(\mathrm{Id},0\right)}{\to }Y×I$ for $I=\left[0,1\right]$ the standard interval and every commuting diagram

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

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

$\begin{array}{ccc}Y& \to & E\\ ↓& {}^{\sigma }↗& ↓\\ Y×I& \to & B\end{array}$\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 ${E}^{I}\to {B}^{I}{×}_{B}E$ induced by the commuting square

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

${X}^{\Delta \left[1\right]}\to {X}^{\left\{0\right\}}{×}_{{S}^{\left\{0\right\}}}{S}^{{\Delta }^{1}}$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 $\infty$-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

$\begin{array}{ccc}{X}_{c}& \to & X\\ ↓& & ↓\\ \left\{c\right\}& \to & C\end{array}$\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 $\left(\infty ,1\right)$-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 $\mathrm{LAn}=\mathrm{LLP}\left(\mathrm{RLP}\left({\mathrm{LAn}}_{0}\right)\right)$ for the following choices of ${\mathrm{LAn}}_{0}$:

${\mathrm{LAn}}_{0}=$

1. the collection of all left horn inclusions

$\left\{\Lambda \left[n{\right]}_{i}\to \Delta \left[n\right]\mid 0\le i;

1. the collection of all inclusions of the form

$\left(\Delta \left[m\right]×\left\{0\right\}\right)\coprod _{\partial \Delta \left[m\right]×\left\{0\right\}}\left(\partial \Delta \left[m\right]×\Delta \left[1\right]\right)↪\Delta \left[m\right]×\Delta \left[1\right]$(\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

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

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

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

###### 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

$\left(A×B\prime \right)\coprod _{A×B}\left(A\prime ×B\right)\to A\prime ×B\prime$(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

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

$\left({A}_{0}\star B\right)\coprod _{{A}_{0}\star {B}_{0}}\left(A\star {B}_{0}\right)\to A\star B$(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 $X×\Delta \left[1\right]\to S$ from $s\circ p$ to ${\mathrm{Id}}_{X}$ then $s$ is left anodyne.

This appears as HTT, prop. 2.1.2.11.

Revised on June 17, 2012 15:14:41 by Stephan Alexander Spahn (85.0.152.155)