nLab
Kan complex

Idea

A Kan complex is a geometric model of an $\mathrm{\infty }$-groupoid based on the shape modeled by the simplex category.

Definition

A Kan complex is a simplicial set $S$ that satisfies the Kan condition,

• which says that all horns of the simplicial set have fillers,

• which means equivalently that the unique morphism $S\to \mathrm{pt}$ from $S$ to the point is a Kan fibration,

• which means equivalently that for all diagrams

  $$\begin{array}{ccc}{\Lambda }^i[n]& \to & S\\ \downarrow & & \downarrow \\ \Delta [n]& \to & \mathrm{pt}\end{array}\leftrightarrow \begin{array}{ccc}{\Lambda }^i[n]& \to & S\\ \downarrow & & \\ \Delta [n]\end{array}$$\array{ \Lambda^i[n] &\to& S \\ \downarrow && \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow && \\ \Delta[n] }

  there exists a diagonal morphism

  $$\begin{array}{ccc}{\Lambda }^i[n]& \to & S\\ \downarrow & \nearrow & \downarrow \\ \Delta [n]& \to & \mathrm{pt}\end{array}\leftrightarrow \begin{array}{ccc}{\Lambda }^i[n]& \to & S\\ \downarrow & \nearrow & \\ \Delta [n]\end{array}.$$\array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \downarrow \\ \Delta[n] &\to& pt } \;\;\; \leftrightarrow \;\;\; \array{ \Lambda^i[n] &\to& S \\ \downarrow &\nearrow& \\ \Delta[n] } \,.

• This in turn means equivalently that the map from $n$-simplices to $(n,i)$-horns is an epimorphism

  $$[\Delta [n],S]\to >[{\Lambda }^i[n],S].$$[\Delta[n], S] \to\gt [\Lambda^i[n],S] \,.

  In this last form the Kan condition is useful for defining internal Kan complexes: for instance a smooth Kan complex can be defined as a simplicial object in Diff such that the morphisms $[\Delta [n],S]\to [{\Lambda }^i[n],S]$ are surjective submersions.

Remarks

• Kan complexes are among the most convenient and popular models for ∞-groupoids. The horn filling condition from this point of view is read as guaranteeing that

  • for all collection of $(n-1)$ composable $n$-cells (a horn ${\Lambda }^k[n]$) there exists an $n$-cell – their composite – and an $(n-1)$-cell connecting the original $(n-1)$ $n$-cells with their composite. Depending on $k$, this interpretation in terms of composition implies that one thinks of all cells as being reversible. Therefore this models an ∞-groupoid.

  For illustrations of the horn-filler conditions see Kan fibration.

• Whatever other definition of ∞-groupoid one considers, it is expected to map to a Kan complex under the nerve.

• A slight weakening of the Kan condition, the weak Kan condition leads to the definition of quasi-category.

• Kan complexes are the fibrant objects in the model structures on simplicial sets for which fibrations are Kan fibrations.

  In this context, a weak equivalence between Kan complexes is a morphism of simplicial sets that induces an isomorphism on the simplicial homotopy groups of the two Kan complexes.

Examples

Kan complexes from nerves of $n$-groupoids

Proposition

The nerve $N(C)$ of a small category is a Kan complex if and only if $C$ is a groupoid.

The existence of inverse morphisms in $D$ corresponds to the fact that in the Kan complex $N(D)$ the “outer” horns

$$\begin{array}{ccc}& & d_0\\ & & & {\searrow }^f\\ d_1& & \stackrel{{\mathrm{Id}}_{d_1}}{\to }& & d_1\end{array}\mathrm{and}\begin{array}{ccc}& & d_1\\ & {}^f\nearrow & & \\ d_0& & \stackrel{{\mathrm{Id}}_{d_0}}{\to }& & d_0\end{array}$$\array{ && d_0 \\ & && \searrow^{f} \\ d_1 &&\stackrel{Id_{d_1}}{\to} && d_1 } \;\;\; \;\;\; and \;\;\; \;\;\; \array{ && d_1 \\ & {}^f\nearrow && \\ d_0 &&\stackrel{Id_{d_0}}{\to} && d_0 }

have fillers

$$\begin{array}{ccc}& & d_0\\ & {}^{f^{-1}}\nearrow & & {\searrow }^f\\ d_1& & \stackrel{{\mathrm{Id}}_{d_1}}{\to }& & d_1\end{array}\mathrm{and}\begin{array}{ccc}& & d_1\\ & {}^f\nearrow & & {\searrow }^{f^{-1}}\\ d_0& & \stackrel{{\mathrm{Id}}_{d_0}}{\to }& & d_0\end{array}$$\array{ && d_0 \\ & {}^{f^{-1}}\nearrow&& \searrow^{f} \\ d_1 &&\stackrel{Id_{d_1}}{\to} && d_1 } \;\;\; \;\;\; and \;\;\; \;\;\; \array{ && d_1 \\ & {}^f\nearrow && \searrow^{f^{-1}} \\ d_0 &&\stackrel{Id_{d_0}}{\to} && d_0 }

(even unique fillers, due to the properties of the nerve of an ordinary category).

This is one way to see and motivate that a simplicial set that is a Kan complex but which does not necessarily have unique fillers makes models an ∞-groupoid.

Accordingly

Proposition

The nerve $N(C)$ of a strict ω-category is a Kan complex if and only if $C$ is a strict ω-groupoid.

singular simplicial complexes / fundamental $\mathrm{\infty }$-groupoids

For $X$ a topological space, its singular simplicial complex is the simplicial set $\Pi (X)$ (often denoted $S(X)$) whose set of $n$-simplices is the hom-set

$$\Pi (X{)}_n:=\mathrm{Top}({\Delta }_{\mathrm{Top}}^n,X)$$\Pi(X)_n := Top(\Delta^n_{Top}, X)

in Top of continuous maps from the standard topological $n$-simplex ${\Delta }_{\mathrm{Top}}^n$ into $X$.

Using the fact that the ${\Delta }_{\mathrm{Top}}^n$ arrange themselves into a cosimplicial space

$${\Delta }_{\mathrm{Top}}:\Delta \to \mathrm{Top}$$\Delta_{Top} : \Delta \to Top

in the obvious way, the $(\Pi (X{)}_n)$ become a simplicial set in the corresponding obvious way. For instance the face maps are induced by restricting maps to $X$ along the face inclusions ${\delta }^i:{\Delta }^{n-1}\hookrightarrow {\Delta }^n$.

That $\Pi (X)$ is indeed a Kan complex is intuitively clear. Technically it follows from the fact that the inclusions ${{{\Lambda }^n}_{\mathrm{Top}}}_k\hookrightarrow {\Delta }_{\mathrm{Top}}^n$ of topological horns into topological simplices are retracts, in that there are continuous maps ${\Delta }_{\mathrm{Top}}^n\to {{{\Lambda }^n}_{\mathrm{Top}}}_k$ given by “squashing” a topological $n$-simplex onto parts of its boundary, such that

$$({{{\Lambda }^n}_{\mathrm{Top}}}_k\to {\Delta }_{\mathrm{Top}}^n\to {{{\Lambda }^n}_{\mathrm{Top}}}_k)=\mathrm{Id}.$$({{\Lambda^n}_{Top}}_k \to \Delta^n_{Top} \to {{\Lambda^n}_{Top}}_k) = Id \,.

Therefore the map $[{\Delta }^n,\Pi (X)]\to [{\Lambda }_k^n,\Pi (X)]$ is an epimorphism, since it is equal to to $\mathrm{Top}({\Delta }^n,X)\to \mathrm{Top}({\Lambda }_k^n,X)$ which has a right inverse $\mathrm{Top}({\Lambda }_k^n,X)\to \mathrm{Top}({\Delta }^n,X)$.

The ∞-groupoid represented by the Kan complex $\Pi (X)$ is called the fundamental ∞-groupoid of $X$.

This example is the universal one: up to weak equivalence of Kan complexes every Kan complex is the fundamental $\mathrm{\infty }$-groupoid of a (compactly generated, weakly Hausdorff) topological space.

This is the statement of the homotopy hypothesis (which is a theorem for $\mathrm{\infty }$-groupoids modeled as Kan complexes.