Kan complex

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 $\mathrm{Sing}(T)$ 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 of a groupoid 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.