---
tag: H2YS8SX31W
kind: definition
name: Kan complex
msc: [18N50, 55U10]
defines: FDFYCM3YYC
uses: [WW81ARB8K9, RWBMXQW6S5, RV0MKJ3CQ6, FH3C0T643F]
sources:
  - url: https://en.wikipedia.org/wiki/Kan_fibration
    locator: Definitions
---

## Definition

For $n \ge 1$ and $0 \le k \le n$, the *horn* $\Lambda^n_k$ is the simplicial subset of the standard $n$-simplex $\Delta^n$ generated by all faces $d_i \Delta^n$ with $i \ne k$.
A [[simplicial set|WW81ARB8K9]] $X$ is a *Kan complex* if for all $n \ge 1$ and $0 \le k \le n$, every map of simplicial sets $\Lambda^n_k \to X$ extends to a map $\Delta^n \to X$.

## Remarks

Equivalently, $X$ is a Kan complex when the map $X \to \Delta^0$ is a Kan fibration.
A map $\Lambda^n_k \to X$ is the same as a list of simplices $x_i \in X_{n-1}$ for $i \ne k$ with $d_i x_j = d_{j-1} x_i$ whenever $i < j$ and $i, j \ne k$.
A filler is an $x \in X_n$ with $d_i x = x_i$ for $i \ne k$.

Example: for a topological space $T$, the [[singular simplicial set|RWBMXQW6S5]] $\mathrm{Sing}(T)$ is a Kan complex.
A map $\Lambda^n_k \to \mathrm{Sing}(T)$ is a continuous map from the geometric horn $\abs{\Lambda^n_k}$ to $T$, and $\abs{\Lambda^n_k}$ is a retract of the topological simplex $\abs{\Delta^n}$, so the map extends.

Example: the [[nerve|RV0MKJ3CQ6]] of a [[groupoid|FH3C0T643F]] is a Kan complex.

Non-example: $\Delta^1$, the nerve of the poset $0 < 1$, is not a Kan complex.
The map $\Lambda^2_0 \to \Delta^1$ sending the vertices $0, 1, 2$ to $0, 1, 0$ is defined, since the edges $0 \to 1$ and $0 \to 2$ go to $0 \le 1$ and $0 \le 0$.
A filler would be an order-preserving map $[2] \to [1]$ with values $0, 1, 0$, which does not exist.
More generally, the nerve of a category is a Kan complex if and only if the category is a groupoid.