nLab Kan fibration

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

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

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

corresponding to the lifting diagram

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

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

This corresponds to the lifting diagram

$\array{ \Lambda_2^1 &\stackrel{y}{\to}& Y \\ \downarrow &{}^{\hat h}\nearrow& \downarrow^\pi \\ \Delta^2 &\stackrel{h}{\to}& X } \,.$
- Crucial is this condition for the outer horns $\Lambda^n_0$ and $\Lambda^n_n$ , 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 $\Lambda^2_2$ the picture is
  $\array{ &&&&& y_2 \\ &&&& && \searrow \\ Y &\ni& & y_1 &&\to&& y_3 \\ \downarrow^\pi \\ X &\ni& &&& \pi(y_2) \\ &&& & \nearrow &\Downarrow^{\forall h}& \searrow \\ &&& \pi(y_1) &&\to&& \pi(y_2) } \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \array{ &&&&& y_2 \\ &&&& {}^\exists\nearrow &\Downarrow^{\exists \hat h} & \searrow \\ Y &\ni& & y_1 &&\stackrel{\exists}{\to}&& y_3 \\ \downarrow^\pi \\ X &\ni& &&& \pi(y_2) \\ &&& & \nearrow &\Downarrow^{h = \pi(\hat h)}& \searrow \\ &&& \pi(y_1) &&\to&& \pi(y_2) }$
  
  This corresponds to the lifting diagram
  
  $\array{ \Lambda_2^2 &\stackrel{y}{\to}& Y \\ \downarrow &{}^{\hat h}\nearrow& \downarrow^\pi \\ \Delta^2 &\stackrel{h}{\to}& X } \,.$

Variants

Minimal Kan fibration

A Kan fibration $p : E \to B$ is called a minimal Kan fibration if for all cells $x,y : \Delta[n] \to E$ the condition $p(x) = p(y)$ and $\partial_i x = \partial_i y$ implies for all $k$ that $\partial_k x = \partial_k y$ .

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: $\Lambda^n_k$ with $0 \lt k \lt (n-1)$ , not the outer horns $\Lambda^n_0$ and $\Lambda^n_n$ .

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.

Examples

Example

(empty bundle is Kan fibration) Every empty bundle $\varnothing \to X$ is a Kan fibration, since none of the commuting squares that one would have to lift in actually exist

\array{ \Lambda_k^n &\overset{ \not \exists }{\longrightarrow}& \varnothing \\ \big\downarrow && \big\downarrow \\ \Delta^n &\longrightarrow& B }

(keeping in mind that the 0-simplex has no horns, hence that all horns are inhabited, so that there is no morphism from any horn to the empty simplicial set, this being a strict initial object).

Properties

Surjective Kan fibrations

Notice that a Kan fibration need not be a surjection in any sense:

Example

For $X \,\in\, sSet$ any simplicial set, the unique morphism from the initial simplicial set (which is the empty set in each degree) is a Kan fibration (the empty bundle):

\varnothing \underset{\in KanFib}{\longrightarrow} X \,.

This is due to the fact that the 0-simplex has no horns, namely that all horns appearing in the Kan condition (1) are inhabited, so that none of them has any morphism to $\varnothing$ (which is a strict initial object), so that the horn filler condition for the empty bundle is vacuous and hence satisfied.

But:

Lemma

As soon as a Kan fibration is a surjection in degree 0, then it is a surjection in all degrees (and hence an epimorphism of simplicial sets):

X \overset{f}{\underset{\in KanFib}{\longrightarrow}} Y \;\;\text{and}\;\; f_0 \; \text{is surjective} \;\;\;\;\;\; \Rightarrow \;\;\;\;\;\; \underset{n \in \mathbb{N}}{\forall} \;\; f_n \; \text{is surjective}

Proof

Let $\Delta[n] \xrightarrow{\sigma_n} Y$ be any $n$ -simplex (seen, under the Yoneda lemma, as a morphism from the standard $n$ -simplex). We equivalently need to show that this lifts through $f$ , in that we have a commuting diagram of this form:

\array{ && X \\ & \mathllap{{}^{\widehat{\sigma_n}}}\nearrow & \big\downarrow \mathrlap{ {}^{f} } \\ \Delta[n] &\underset{\sigma_n}{\longrightarrow}& Y \mathrlap{\,.} }

Now, by the assumption that $f_0$ is surjective, we do have such a lift at least for any vertex $v$ of $\sigma_n$ , hence we have a commuting diagram of this form:

\array{ \Delta[0] &\overset{\widehat{v}}{\longrightarrow}& X \\ \big\downarrow && \big\downarrow\mathrlap{{}^{f}} \\ \Delta[n] &\underset{\sigma_n}{\longrightarrow}& Y \mathrlap{\,.} }

But the left morphism is an acyclic cofibration in the classical model structure on simplicial sets. These have the left lifting property against Kan fibrations, meaning that this last square has a lift $\widehat{\sigma_n}$ . This exhibits the desired preimage of $\sigma_n$ .

In fact:

Lemma

As soon as a Kan fibration $f$ is a surjection on connected components, $\pi_0(f) \colon \pi_0(X_\bullet) \twoheadrightarrow \pi_0(Y_\bullet)$ , then it is a surjection in all degrees (and hence an epimorphism of simplicial sets):

Proof

By definition (of simplicial homotopy groups), all vertices in a connected component are connected by some zig-zag $Z$ of edges. But the inclusion of an endpoint vertex $\Delta[0] \hookrightarrow Z$ into (the abstract shape of) such a zig-zag is evidently an acyclic cofibration, hence has the left lifting property against the given Kan fibration (by the classical model structure on simplicial sets). Evaluating this lift at the other endpoint shows that the Kan fibration is surjective on vertices as soon as it is surjective on connected components. Therefore the claim follows with Lemma .

In summary:

Proposition

Let $X \xrightarrow{f} Y$ be any morphism of simplicial sets which is a surjection on connected components: $\pi_0(f) \,\colon\, \pi_0(X) \twoheadrightarrow \pi_0(Y)$ (hence presenting an effective epimorphism of $\infty$ -groupoids, by this Prop.). Then $f$ factors as a simplicial weak equivalence followed by a Kan fibration which is a degreewise surjection (and hence an epimorphism of simplicial sets).

Proof

The factorization follows by standard constructions such as the factorization lemma or the existence of the classical model structure on simplicial sets. With this the claim follows by Lemma .

Remark

The assumption in Prop. is met in particular for acyclic Kan fibrations. In this special case the above factorization is trivial in that acyclic Kan fibrations are already degreewise surjective themselves (see the discussion there).

Acyclic Kan fibrations and weak homotopy equivalences

Theorem

The acyclic Kan fibrations morphisms $f : X \to Y$ 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 $\partial \Delta^n \hookrightarrow \Delta^n$ :

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

e.g. (Goerss-Jardine, chapter I)

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.

Proposition

A Kan fibration $f\colon X\to Y$ is acyclic precisely if the fiber $f^{-1}(y)$ over each vertex $y$ is contractible.

Purely combinatorial proofs of this statement include (Joyal 2008, prop. 8.23, Riehl-Verity 13, lemma 5.4.16)

Pullback and homotopy pullback

Lemma

Let $p \colon X \longrightarrow Y$ be a Kan fibration, def. , and let $f_1,f_2 \colon A \longrightarrow X$ be two morphisms. If there is a left homotopy $f_1 \rightarrow f_2$ between these maps, then there is a fiberwise homotopy equivalence, between the pullback fibrations $f_1^\ast X \simeq f_2^\ast X$ .

(e.g. Goerss-Jardine 96, chapter I, lemma 10.6)

On nerves of groupoids

Theorem

A functor $F \colon C \to D$ between groupoids is k-surjective for all $k$ and hence a surjective equivalence of categories precisely if under the nerve $N(F) : N(C) \to N(D)$ it induces an acyclic fibration of Kan complexes;

Proof

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

\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 $n$ :

$n= 0$ . 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 $N(D)$ lifts through $N(F)$ .

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

This precisely says that $N(F)$ is surjective on 0-cells and hence that $F$ is surjective on objects.
$n=1$ . 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 $a,b \in Obj(C)$ all elements in $Hom(F(a),F(b))$ are hit. Hence that $F$ is a full functor.
$n=2$ . 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 $\partial \Delta[2] \to N(C)$ 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 $a,b \in Obj(C)$ the map $F_{a,b} : Hom(a,b) \to Hom(F(a),F(b))$ is injective. Hence that $F$ is a faithful functor.

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

Universal Kan fibration

See at universal Kan fibration.

Relation to other concepts

Kan fibrations and quasi-fibrations are fibrations in two common model structures on simplicial sets.
Recall that the horn $\Lambda^k[n]$ is the boundary of the $n$ -simplex $\Delta^n$ 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 $X$ for which the unique morphism $X \to pt$ to the terminal simplicial set is a Kan fibration is called a Kan complex.
A simplicial set $X$ for which the unique morphism $X \to pt$ 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 $f : G\to H$ of simplicial groups for which in each dimension, $n$ , the homomorphism $f_n : G_n \to H_n$ 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.

Related concepts

Kan fibration, anodyne morphism
- acyclic Kan fibration
Kan complex
right/left Kan fibration, right/left anodyne map
inner fibration
Cartesian fibration
simplicial homotopy theory
homotopy Kan fibration

References

The original definition is due to Daniel M. Kan, see Definition 3.1 in

Daniel M. Kan, A combinatorial definition of homotopy groups, The Annals of Mathematics 67:2 (1958), 282–312. doi.

This was extended to arbitrary simplicial objects via the Yoneda embedding in Definition 3.2 of

Daniel M. Kan, On c.s.s. categories, Boletín de la Sociedad Matemática Mexicana 2 (1957), 82–94. PDF.

A standard textbook account is

Paul Goerss, Rick Jardine, Simplicial homotopy theory, 1996

That geometric realization takes Kan fibrations to Serre fibrations is due to

Dan Quillen, The geometric realization of a Kan fibration is a Serre fibration, Proc. AMS 19 (1968), 1499–1500