nLab
right/left Kan fibration

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. Cartesian fibrations

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

rspectively.

Definition

A morphism of simplicial sets f:XSf : X \to S is a left fibration or left Kan fibration if it has the right lifting property with respect to all horn inclusions Λ[n] kΔ[n]\Lambda[n]_k \to \Delta[n] except possibly the right outer ones: 0k<n0 \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<kn0 \lt k \leq n.

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

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

for nn \in \mathbb{N} and 0k<n0 \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)sSet/S RFib(S) \subset sSet/S

for the full SSet-subcategory of the overcategory of sSet over SS 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:CatsSetN : Cat \to sSet be the nerve functor and for p:EBp : E \to B a morphism in Cat (a functor), let N(p):N(E)N(B)N(p) : N(E) \to N(B) be the corresponding morphism in sSet.

Then

Proposition

The functor p:EBp : E \to B is an fibration in groupoids precisely if the morphism N(p):N(E)N(B)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 CC a category, the nerve N(C)N(C) is 2-coskeletal. In particular all nn-spheres for n3n \geq 3 have unique fillers

Δ[n] N(C) ! Δ[n](n3) \array{ \partial \Delta[n] &\stackrel{\forall}{\to}& N(C) \\ \downarrow & \nearrow_{\mathrlap{\exists !}} \\ \Delta[n] } \;\;\;\;\; (n \geq 3)

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

Λ[n] N(C) Δ[n](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:EBp : E \to B is an ordinary functor, then N(f):N(E)N(B)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 Λ[n] iΔ[n]\Lambda[n]_i \hookrightarrow \Delta[n] for 0<i<n0 \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)N(B)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

{Λ[n] nΔ[n]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=1n=1 – The existence of all fillers

    Λ[1] 1=Δ {1} e N(E) f^ N(p) Δ {01} f N(B) \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 eEe \in E and morphism f:bp(e)f : b \to p(e) in BB, there exists a morphism f^:b^e\hat f : \hat b \to e in EE such that p(f^)=fp(\hat f) = f.

  • n=2n=2 – The existence of fillers

    Λ[2] 2 e N(E) f^ N(p) Δ[2] f N(B) \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

    e 1 ϵ 12 e 0 ϵ 02 e 2 \array{ && e_1 \\ &&& \searrow^{\mathrlap{\epsilon_{12}}} \\ e_0 &&\stackrel{\epsilon_{02}}{\to}&& e_2 }

    in EE and commuting triangles

    p(e 1) f p(ϵ 12) p(e 0) p(ϵ 02) p(e 2) \array{ && p(e_1) \\ & {}^{\mathllap{f}}\nearrow&& \searrow^{\mathrlap{p(\epsilon_{12}})} \\ p(e_0) &&\stackrel{p(\epsilon_{02})}{\to}&& p(e_2) }

    in BB, there is a commuting triangle

    e 1 f^ ϵ 12 e 0 ϵ 02 e 2 \array{ && e_1 \\ &{}^{\mathllap{\hat f}}\nearrow&& \searrow^{\mathrlap{\epsilon_{12}}} \\ e_0 &&\stackrel{\epsilon_{02}}{\to}&& e_2 }

    in EE, such that p(f^)=fp(\hat f ) = f.

  • n=3n=3

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

    Suppose in the above situation two lifts (f^) 1(\hat f)_1 and (f^) 2(\hat f)_2 are found. Together these yield a Λ[3] 3\Lambda[3]_3-horn in N(E)N(E). The filler condition says this can be filled, which implies that (f^) 1=(f^) 2(\hat f)_1 = (\hat f)_2.

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

    By similar reasoning one sees that this is all the n=3n=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:CDp : C \to D is a left fibration then p op:C opD opp^{op} : C^{op} \to D^{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:EBp : 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(Id,0)Y×IY \stackrel{(Id, 0)}{\to} Y \times I for I=[0,1]I = [0,1] the standard interval and every commuting diagram

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

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

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

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

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

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

Proposition

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

X Δ[1]X {0}× S {0}S Δ 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 TT, left fibrations STS \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*C \to * a right (left) fibration over the point, CC 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 cX_c of every right fibration XCX \to C over every point cCc \in C, i.e. the pullback

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

is a Kan complex.

Proposition

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

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 (,1)(\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:KCp : K \to C be an arbitrary morphism to a quasi-category CC and let C p/C_{p/} be the corresponding under quasi-category. Then the canonical propjection C p/CC_{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))LAn = LLP(RLP(LAn_0)) for the following choices of LAn 0LAn_0:

LAn 0=LAn_0 =

  1. the collection of all left horn inclusions

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

  1. the collection of all inclusions of the form

    (Δ[m]×{0}) Δ[m]×{0}(Δ[m]×Δ[1])Δ[m]×Δ[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×{0}) S×{0}(S×Δ[1])S×Δ[1] (S' \times \{0\}) \coprod_{S \times \{0\}} (S \times \Delta[1]) \hookrightarrow S' \times \Delta[1]

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

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

Proof

Corollary

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

(A×B) A×B(A×B)A×B (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:XSp : X \to S a left fibration and i:ABi : A \to B a cofibration of simplicial sets, the canonical morphism

q:X BX A× S AS B q : X^B \to X^A \times_{S^A} S^B

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

This appears as HTT, cor. 2.1.2.9.

Proposition

For f:A 0Af : A_0 \to A and g:B 0Bg : B_0 \to B two inclusions of simplicial sets with ff left anodyne, we have that the canonical morphism

(A 0B) A 0B 0(AB 0)AB (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:BSp : B \to S be a morphism of simplicial sets and i:ABi : A \to B a left anodyne morphism, then the restriction morphism of under quasi-categories

S /pS /p 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:XSp : X \to S be a morphism of simplicial sets with section s:SXs : S \to X. If there is a fiberwise simplicial homotopy X×Δ[1]SX \times \Delta[1] \to S from sps \circ p to Id XId_X then ss 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)