nLab
Kan complex

Idea

The notion of a Kan complex is an abstraction of the structure found in the singular simplicial complex of a topological space.

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

A Kan complex is a collection of $k$-simplex-shaped k-morphisms for all $k\in \mathbb{N}$, such that for all composable $k$-morphisms a composite does exist (not necessarily uniquely) and such that all $k$-morphisms are invertible under this composition.

In a Kan complex there are no specified composites. But one may choose out for all composable $k$-morphisms a composite and thus arrive at an algebraic model for $\mathrm{\infty }$-groupoids: algebraic Kan complexes.

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

  $$[\Delta [n],S]\twoheadrightarrow [{\Lambda }^i[n],S].$$[\Delta[n], S]\, \twoheadrightarrow \,[\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.

How to think of Kan complexes as $\mathrm{\infty }$-groupoids

We expand here a bit on how a Kan complex may naturally be thought of as an ∞-groupoid: a higher category in which all k-morphisms for all $k\in \mathbb{N}$ are invertible.

For this interpretation, one thinks of a $k$-dimensional cell $\varphi \in C_K$ of a Kan complex $C$ as a k-morphism whose

• source is the $(k-1)$-morphism given by the pasting diagram of the $(k-1)$-cells that are the faces $d_k\varphi ,d_{k-2}\varphi ,d_{k-4}\varphi \cdots$ of $\varphi$;

• target is the $(k-1)$-morphism given by the pasting diagram of the $(k-1)$-cells that are the faces $d_{k-1}\varphi ,d_{k-3}\varphi ,d_{k-5}\varphi \cdots$ of $\varphi$

where $d_i:C_k\to C_{k-1}$ are the face maps of the simplicial set $C$.

This is easy to see in low dimensions:

• a 1-cell $\varphi \in C_1$ in the simplicial set $C$ has a single source 0-cell $x:=d_1\varphi$ and a single target 0-cell $y:=d_0\varphi$ and hence may be pictured as a morphism

  $$x\stackrel{\varphi }{\to }y.$$x \stackrel{\phi}{\to} y \,.

• a 2-cell $\varphi \in C_2$ in the simplicial set $C$ has two incoming 1-cells $d_2\varphi ,d_0\varphi \in C_1$ and one outgoing 1-cell $d_1\varphi \in C_1$, and if we think of the two incoming 1-cells as representing the composite of the corresponding 1-morphisms, we may picture te 2-cell $\varphi$ here as a globular 2-morphism

  $$\begin{array}{ccc}& & x_1\\ & {}^{d_2\varphi }\nearrow & {\Downarrow }^{\varphi }& {\searrow }^{d_0\varphi }\\ x_0& & \underset{d_1\varphi }{\to }& & x_2\end{array}.$$\array{ && x_1 \\ & {}^{\mathllap{d_2 \phi}}\nearrow &\Downarrow^\phi& \searrow^{\mathrlap{d_0 \phi}} \\ x_0 &&\underset{d_1 \phi}{\to}&& x_2 } \,.

More in detail, one may think of the incoming two adjacent $1$-cells here as not being the composite of these two morphism, but just as a composable pair, and should think of the existence of the 2-morphism $\varphi$ here as being a compositor in a bicategory that shows how the composable pair is composed to the morphism $d_1\varphi$.

So if an $\mathrm{\infty }$-groupoid is thought of as a globular ω-category in which all k-morphisms are invertible, then the corresponding Kan complex is the nerve or rather the ω-nerve of this ω-category.

Notably if $C$ is to be regarded as (the nerve of) an ordinary groupoid, every composable pair of morphisms has a unique composite, and hence there should be a unique 2-cell

$$\begin{array}{ccc}& & x_1\\ & {}^f\nearrow & \Downarrow & {\searrow }^g\\ x_0& & \underset{h=g\circ f}{\to }& & x_2\end{array}$$\array{ && x_1 \\ & {}^{f}\nearrow &\Downarrow& \searrow^{g} \\ x_0 &&\underset{h = g \circ f}{\to}&& x_2 }

that is the unique identity 2-morphism

$$g\circ f\stackrel{=}{\Rightarrow }h.$$g \circ f \stackrel{=}{\Rightarrow} h \,.

More generally, in a 2-groupoid there may be non-identity 2-morphisms, and hence for any 1-morphism $k_x{}_0\to x_2$ 2-isomorphic to $h$, there may be many 2-morphisms $g\circ f\Rightarrow k$, hence many 2-cells

$$\begin{array}{ccc}& & x_1\\ & {}^f\nearrow & {\Downarrow }^{\simeq }& {\searrow }^g\\ x_0& & \underset{k}{\to }& & x_2\end{array}.$$\array{ && x_1 \\ & {}^{f}\nearrow &\Downarrow^{\simeq}& \searrow^{g} \\ x_0 &&\underset{k }{\to}&& x_2 } \,.

All we can say for sure is that at least one such 2-cell exists, and that the 2-cells themselves may be composed in some way. This is precisely what the horn-filler conditions in a Kan complex encode.

To see this for the composition of 2-morphisms, we need to move further up in dimension. The task of analyzing the combinatorics of k-simplices and their boundaries in the above fashion quickly goes beyond what can be handled in a naive fashion. Luckily, this combinatorial problem has been completely solved by Ross Street in his work on orientals.

The $k$-oriental $O(k)$ is precisely the prescription for how exactly to think of a $k$-simplex as being a k-morphism in an omega-category. The first few look like this:

$$\begin{array}{cc}O({\Delta }^0)=& \{0\}\\ O({\Delta }^1)=& \{0\to 1\}\\ O({\Delta }^2)=& \left\{\begin{array}{c} $ 1$ $ 0$ $ 2$ \end{array}\right\}\\ O({\Delta }^3)=& \left\{\begin{array}{c} $ 1$ $ 0$ $ 2$ $ 3$ \end{array}\right\}\\ O({\Delta }^4)=& \left\{\begin{array}{c} $ 2$ $ 1$ $ 3$ $ 0$ $ 4$ \end{array}\right\}\end{array}$$\array{\arrayopts{\rowalign{center}} O(\Delta^0) = & \{ 0\} \\ O(\Delta^1) = & \left\{ 0 \to 1\right\} \\ O(\Delta^2) = & \left\{ \array{\begin{svg} <svg xmlns="http://www.w3.org/2000/svg" width="6em" height="4em" viewBox="0 0 60 40"> <defs> <marker id="svg295arrowhead" viewBox="0 0 10 10" refX="0" refY="5" markerUnits="strokeWidth" markerWidth="8" markerHeight="5" orient="auto"> <path d="M 0 0 L 10 5 L 0 10 z"/> </marker> <marker id="svg296arrowhead" viewBox="0 0 10 10" refX="0" refY="5" markerUnits="strokeWidth" markerWidth="4" markerHeight="2.5" orient="auto"> <path d="M 0 0 L 10 5 L 0 10 z"/> </marker> </defs> <g font-size="10"> <foreignObject x="25" 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In fact, the omega-nerve $N(K)$ of an omega-category $K$ is the simplicial set whose collection of $k$-cells $N(K{)}_k:=\mathrm{Hom}(O(k),K)$ is precisely the collection of images of the $k$th oriental $O(k)$ in $K$.

This is fully formally the prescription of how to think of a Kan complex as an $\mathrm{\infty }$-groupoid: the Kan complex $C$ is the omega-nerve of an omega-category in which all morphism are invertible:

• the $k$-cells in $C_k$ are precisely the collection of $k$-morphisms ihn the omega-category of shape the $k$th oriental $O(k)$;

• the horn-filler conditions satisfied by these cells is precisely a reflection of the fact that

  1. there exists a notion of composition of adjacent k-morphisms in the omega-category;

  2. under this composition all $k$-morphisms have an inverse.

We have already seen in low dimension how the existence of composites in an $\omega$-category is reflected in the fact that in a Kan-complex certain 22-simplices exist, and how the non-uniqueness of these 2-simplices reflects the existence of nontrivial 2-morphisms.

To see in a similar fashion that the Kan condition ensures the existence of inverses consider an outer horn in $C$, a diagram of 1-cells of the form

$$\begin{array}{ccc}& & x_1\\ & {}^f\nearrow \\ x_0& & \underset{h}{\to }& & x_2\end{array}.$$\array{ && x_1 \\ & {}^{\mathllap{f}}\nearrow \\ x_0 &&\underset{h}{\to}&& x_2 } \,.

In general given such a diagram in a category, there is no guarantee that the corresponding triangle as above will exist in its nerve. But if the category is a groupoid, then it is guaranteed that the missing 1-face can be chose to be the inverse of $f$ composed with the morphism $h$, and there is at least one 2-morphism

$$\begin{array}{ccc}& & x_1\\ & {}^f\nearrow & {\Downarrow }^{\simeq }& {\searrow }^{h\circ f^{-1}}\\ x_0& & \underset{h}{\to }& & x_2\end{array}.$$\array{ && x_1 \\ & {}^{\mathllap{f}}\nearrow &\Downarrow^{\simeq}& \searrow^{\mathrlap{h \circ f^{-1}}} \\ x_0 &&\underset{h}{\to}&& x_2 } \,.

A similar analysis for higher dimensional cells shows that the fact that a Kan complex has all horn fillers encodes precisely the fact that it is the omega-nerve of an omega-category in which all k-morphisms for all $k$ are composable if adjacent and have a weak inverse.