# nLab right/left Kan fibration

Contents

### Context

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

(∞,1)-category theory

Background

Basic concepts

Universal constructions

Local presentation

Theorems

Extra stuff, structure, properties

Models

# 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

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

The functor $p : E \to B$ is a 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 $n \geq 3$ 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

$\array{ \Lambda_1 = \Delta^{\{1\}} &\stackrel{e}{\to}& N(E) \\ \downarrow & {}^{{\hat f}}\nearrow & \downarrow^{\mathrlap{N(p)}} \\ \Delta^{\{0 \to 1\}} &\stackrel{f}{\to}& N(B) }$

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

$\array{ \Lambda_2 &\stackrel{e}{\to}& N(E) \\ \downarrow & {}^{{\hat f}}\nearrow & \downarrow^{\mathrlap{N(p)}} \\ \Delta &\stackrel{f}{\to}& N(B) }$

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 $\Lambda_3$-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

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

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

###### 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

$(\Delta[m] \times \{0\}) \coprod_{\partial \Delta[m] \times \{0\}} (\partial \Delta[m] \times \Delta) \hookrightarrow \Delta[m] \times \Delta$
2. the collection of all inclusions of the form

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

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

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

###### 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 $X \times \Delta \to S$ from $s \circ p$ to $Id_X$ then $s$ is left anodyne.

This appears as HTT, prop. 2.1.2.11.