nLab
Kan fibration

Contents

Definition

A Kan fibration is one of the notions of fibrations of simplicial sets.

A Kan fibration is a morphism π:YX\pi : Y \to X of simplicial sets with the lifting property for all horn inclusions.

This means that for

Λ k[n] Y π Δ n X \array{ \Lambda^k[n] &\to& Y \\ \downarrow && \downarrow^\pi \\ \Delta^n &\to& X }

a commuting square with n1n\ge 1 and 0kn0\le k\le n, there always exists a lift

Λ k[n] Y π Δ n X. \array{ \Lambda^k[n] &\to& Y \\ \downarrow &\nearrow& \downarrow^\pi \\ \Delta^n &\to& X } \,.

In terms of the canonical powering of simplicial sets over sets, this is equivalent to the morphisms

Y Δ[n]Y Λ k[n]× X Λ k[n]X Δ k[n] Y^{\Delta[n]} \to Y^{\Lambda^k[n]} \times_{X^{\Lambda^k[n]}} X^{\Delta^k[n]}

all being epimorphisms. (Here, for instance, Y Λ k[n]Y^{\Lambda^k[n]} is the set of tuples of (n1)(n-1)-cells in YY that glue along their boundaries to an image of the kkth nn-horn.)

Illustration

Kan fibrations are combinatorial analogs of Serre fibrations of topological spaces. In fact, under the Quillen equivalence of the standard model structure on topological spaces and the standard model structure on simplicial sets, Kan fibrations map to Serre fibrations.

Recall the shape of the horns in low dimension.

  • -n=1n=1- The horns Λ 0 1\Lambda^1_0 and Λ 1 1\Lambda^1_1 of the 1-simplex are just copies of the 0-simplex Δ 0\Delta^0 regarded as the left and right endpoint of Δ 1\Delta^1. For n=1n= 1 the above condition says that for π:YX\pi : Y \to X a Kan fibration we have

    Y y π X π(y) f xY y f^ x^ π X π(y) f=π(f^) x=π(x) \array{ Y &\ni & y \\ \downarrow^\pi \\ X &\ni& \pi(y) &\stackrel{\forall f}{\to}& x } \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \array{ Y &\ni& y &\stackrel{\exists \hat f}{\to}& \exists \hat x \\ \downarrow^\pi \\ X &\ni& \pi(y) &\stackrel{f = \pi(\hat f)}{\to}& x = \pi(x) }

    corresponding to the lifting diagram

    Λ 1 1 y Y f^ π Δ 1 f X. \array{ \Lambda_1^1 &\stackrel{y}{\to}& Y \\ \downarrow &{}^{\hat f}\nearrow& \downarrow^\pi \\ \Delta^1 &\stackrel{f}{\to}& X } \,.
  • -n=2n=2- the horn Λ 1 2\Lambda^2_1 consists of the two top sides of a triangle. For this the Kan condition says that for any two composable 1-cells in YY that have a “composite up to a 2-cell” in XX, there exists a corresponding “composite up to a 2-cell” in YY that projects down to the one in XX:

    y 2 Y y 1 y 3 π X π(y 2) h π(y 1) π(y 2) y 2 h^ Y y 1 y 3 π X π(y 2) h=π(h^) π(y 1) π(y 2) \array{ &&&&& y_2 \\ &&&& \nearrow && \searrow \\ Y &\ni& & y_1 &&&& y_3 \\ \downarrow^\pi \\ X &\ni& &&& \pi(y_2) \\ &&& & \nearrow &\Downarrow^{\forall h}& \searrow \\ &&& \pi(y_1) &&\to&& \pi(y_2) } \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \array{ &&&&& y_2 \\ &&&& \nearrow &\Downarrow^{\exists \hat h}& \searrow \\ Y &\ni& & y_1 &&\stackrel{\exists}{\to}&& y_3 \\ \downarrow^\pi \\ X &\ni& &&& \pi(y_2) \\ &&& & \nearrow &\Downarrow^{h = \pi(\hat h)}& \searrow \\ &&& \pi(y_1) &&\to&& \pi(y_2) }

    This corresponds to the lifting diagram

    Λ 2 1 y Y h^ π Δ 2 h X. \array{ \Lambda_2^1 &\stackrel{y}{\to}& Y \\ \downarrow &{}^{\hat h}\nearrow& \downarrow^\pi \\ \Delta^2 &\stackrel{h}{\to}& X } \,.
    • Crucial is this condition for the outer horns Λ 0 n\Lambda^n_0 and Λ n n\Lambda^n_n, where it says that the above works not only when edges are composable, but also when they meet just at their sources or their targets. For instance for the horn Λ 2 2\Lambda^2_2 the picture is
      y 2 Y y 1 y 3 π X π(y 2) h π(y 1) π(y 2) y 2 h^ Y y 1 y 3 π X π(y 2) h=π(h^) π(y 1) π(y 2) \array{ &&&&& y_2 \\ &&&& && \searrow \\ Y &\ni& & y_1 &&\to&& y_3 \\ \downarrow^\pi \\ X &\ni& &&& \pi(y_2) \\ &&& & \nearrow &\Downarrow^{\forall h}& \searrow \\ &&& \pi(y_1) &&\to&& \pi(y_2) } \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \array{ &&&&& y_2 \\ &&&& {}^\exists\nearrow &\Downarrow^{\exists \hat h} & \searrow \\ Y &\ni& & y_1 &&\stackrel{\exists}{\to}&& y_3 \\ \downarrow^\pi \\ X &\ni& &&& \pi(y_2) \\ &&& & \nearrow &\Downarrow^{h = \pi(\hat h)}& \searrow \\ &&& \pi(y_1) &&\to&& \pi(y_2) }

      This corresponds to the lifting diagram

      Λ 2 2 y Y h^ π Δ 2 h X. \array{ \Lambda_2^2 &\stackrel{y}{\to}& Y \\ \downarrow &{}^{\hat h}\nearrow& \downarrow^\pi \\ \Delta^2 &\stackrel{h}{\to}& X } \,.

Variants

Quasi-fibration

A quasi-fibration or weak Kan fibration or inner Kan fibration of simplicial sets is defined as above, but with the lifting property only imposed in inner horns: Λ k n\Lambda^n_k with 0<k<(n1)0 \lt k \lt (n-1), not the outer horns Λ 0 n\Lambda^n_0 and Λ n n\Lambda^n_n.

This weakened condition then says that composition of cells may be lifted through the quasi-fibration, but not necessarily inversion of 1-cells. See fibrations of quasi-categories for more details.

Left and right Kan fibration

Similarly, a left Kan fibration is one that has the lifting property for all horns except possibly the last one. and a right Kan fibration is one that has the lifting property for all horns except possibly the first one. See fibrations of quasi-categories for more details.

Minimal Kan fibration

A Kan fibration p:EBp : E \to B is called a minimal Kan fibration if for all cells x,y:Δ[n]Ex,y : \Delta[n] \to E the condition p(x)=p(y)p(x) = p(y) and ix= iy\partial_i x = \partial_i y implies for all kk that kx= ky\partial_k x = \partial_k y.

Properties

Theorem

The acyclic Kan fibrations morphisms f:XYf : X \to Y of Kan complexes that are both Kan fibrations as well as weak equivalences in that they induce isomorphisms on all simplicial homotopy groups (i.e. the acyclic fibrations of Kan complexes) are precisely the morphisms that have the right lifting property with respect to all boundary inclusions Δ nΔ n\partial \Delta^n \hookrightarrow \Delta^n:

Δ[n] X f Δ[n] Y. \array{ \partial \Delta[n] &\to& X \\ \downarrow &{}^\exists\nearrow& \downarrow^f \\ \Delta[n] &\to& Y } \,.
Proof

A proof is in chapter I of

  • Goerss-Jardine, Simplicial homotopy theory. Explicitly, it is theorem 7.10 here.
Corollary

Kan fibrations and acyclic Kan fibrations are both stable under pullback.

Proof

Because every class of morphisms defined by a right lifting property is stable under pullback.

Remark

From this it follows readily that Kan complexes form a Brownian category of fibrant objects.

Let C,DC, D be ordinary groupoids and N(C)N(C), N(D)N(D) their ordinary nerves. We’d like to show in detail that

Theorem

A functor F:CDF : C \to D is

Proof

We know that both N(C)N(C) and N(D)N(D) are Kan complexes. By the above theorem it suffices to show that N(f)N(f) being a surjective equivalence is the same as having all lifts

δΔ[n] N(C) N(F) Δ[n] N(D). \array{ \delta \Delta[n] &\to& N(C) \\ \downarrow &{}^\exists\nearrow& \downarrow^{N(F)} \\ \Delta[n] &\to& N(D) } \,.

We check successively what this means for increasing nn:

  • n=0n= 0. In degree 0 the boundary inclusion is that of the empty set into the point *\emptyset \hookrightarrow {*}. The lifting property in this case amounts to saying that every point in N(D)N(D) lifts through N(F)N(F).

    N(C) N(F) * N(D) N(C) N(F) * N(D). \array{ \emptyset &\to& N(C) \\ \downarrow &{}^\exists\nearrow& \downarrow^{N(F)} \\ {*} &\to& N(D) } \Leftrightarrow \array{ && N(C) \\ &{}^\exists\nearrow& \downarrow^{N(F)} \\ {*} &\to& N(D) } \,.

    This precisely says that N(F)N(F) is surjective on 0-cells and hence that FF is surjective on objects.

  • n=1n=1. In degree 1 the boundary inclusion is that of a pair of points as the endpoints of the interval {,}{}\{\circ, \bullet\} \hookrightarrow \{\circ \to \bullet\}. The lifting property here evidently is equivalent to saying that for all objects a,bObj(C)a,b \in Obj(C) all elements in Hom(F(a),F(b))Hom(F(a),F(b)) are hit. Hence that FF is a full functor.

  • n=2n=2. In degree 2 the boundary inclusion is that of the triangle as the boundary of a filled triangle. It is sufficient to restrict attention to the case that the map Δ[2]N(C)\partial \Delta[2] \to N(C) sends the top left edge of the triangle to an identity. Then the lifting property here evidently is equivalent to saying that for all objects a,bObj(C)a,b \in Obj(C) the map F a,b:Hom(a,b)Hom(F(a),F(b))F_{a,b} : Hom(a,b) \to Hom(F(a),F(b)) is injective. Hence that FF is a faithful functor.

    ( a Id a f a g b)N(F)( a Id a = F(f) a F(g) b) \left( \array{ && a \\ & {}^{Id_a}\nearrow && \searrow^{f} \\ a &&\stackrel{g}{\to}&& b } \right) \stackrel{N(F)}{\mapsto} \left( \array{ && a \\ & {}^{Id_a}\nearrow &\Downarrow^=& \searrow^{F(f)} \\ a &&\stackrel{F(g)}{\to}&& b } \right)

Relation to other concepts

  • Kan fibrations and quasi-fibrations are fibrations in two common model structures on simplicial sets.

  • Recall that the horn Λ k[n]\Lambda^k[n] is the boundary of the nn-simplex Δ n\Delta^n with one face removed. If in the above definition one replaces horns with the full boundaries of simplices, one obtaines the definition of a hypercover, the acyclic fibrations in the classical model structure on simplicial sets.

  • A simplicial set XX for which the unique morphism XptX \to pt to the terminal simplicial set is a Kan fibration is called a Kan complex.

  • A simplicial set XX for which the unique morphism XptX \to pt to the terminal simplicial set is a quasi-fibration/weak Kan fibration is called a quasi-category.

  • Just as the underlying simplicial set of a simplicial group is a Kan complex (see algorithm at simplicial group), so also given any simplicial morphism f:GHf : G\to H of simplicial groups for which in each dimension, nn, the homomorphism f n:G nH nf_n : G_n \to H_n is an epimorphism, then the underlying simplicial map of simplicial sets is a Kan fibration. (Apart from a careful choice of section in each dimension, the proof can be constructed from the algorithm given in simplicial group.)

  • A morphism of simplicial sets that has the left lifting property with respect to all Kan fibrations is called an anodyne morphism.

Revised on October 11, 2013 01:05:16 by Anonymous Coward (134.76.82.221)