The standard $2$-simplex $\Delta[2]$ has three horns:

• $\Lambda^2_0$: edges $\{0,1\}$ and $\{0,2\}$ (the two edges meeting at vertex $0$), missing the edge $\{1,2\}$ and the interior. This looks like a "V" opening toward the right.

• $\Lambda^2_1$: edges $\{0,1\}$ and $\{1,2\}$ (the two edges meeting at vertex $1$), missing the edge $\{0,2\}$ and the interior. This is the "inner horn."

• $\Lambda^2_2$: edges $\{0,2\}$ and $\{1,2\}$ (the two edges meeting at vertex $2$), missing the edge $\{0,1\}$ and the interior. This looks like a "V" opening toward the left.

Filling $\Lambda^2_1$ means finding a $2$-simplex with the given edges -- geometrically, filling in a triangle. This is related to composition in a category.

$\Delta[3]$ has four horns $\Lambda^3_0, \Lambda^3_1, \Lambda^3_2, \Lambda^3_3$. Each is obtained by removing one face of the tetrahedron. For instance, $\Lambda^3_1$ consists of the faces $\{0,1,2\}$, $\{0,1,3\}$, and $\{0,2,3\}$ -- everything except the face $\{1,2,3\}$.

Filling a horn in $\Delta[3]$ requires finding a $3$-simplex (a solid tetrahedron) with the given faces. In the context of categories, this encodes associativity of composition.

For any topological space $Y$, the singular simplicial set $\operatorname{Sing}(Y)$ is a Kan complex. Given a horn $\Lambda^n_k \to \operatorname{Sing}(Y)$, which is a continuous map $|\Lambda^n_k| \to Y$, we need to extend it to $|\Delta^n| \to Y$. Since $|\Lambda^n_k|$ is a retract of $|\Delta^n|$ (the topological horn can be deformation-retracted to the boundary missing one face, and then extended), such an extension always exists.

More precisely, there is a retraction $r: |\Delta^n| \to |\Lambda^n_k|$, so $f \circ r$ gives the desired extension.

Consider the category $[1] = (0 \to 1)$ with one non-identity morphism. Its nerve $N([1]) = \Delta[1]$. The horn $\Lambda^2_0 \to \Delta[1]$ given by the identity $\mathrm{id}: 0 \to 0$ on edge $\{0,1\}$ and the morphism $f: 0 \to 1$ on edge $\{0,2\}$ would require a filler -- a $2$-simplex with $d_0$ being a morphism $0 \to 1$ and $d_2$ being $\mathrm{id}_0$. This filler does exist. But the horn $\Lambda^2_2$ with edge $\{0,2\} = f$ and $\{1,2\} = \mathrm{id}_1$ would require finding a morphism $0 \to 1$ composing correctly, which may not exist in general categories.

In fact, $N(\mathcal{C})$ is a Kan complex if and only if $\mathcal{C}$ is a groupoid (every morphism is invertible). The outer horn fillers require inverses.

If $\mathcal{G}$ is a groupoid, then $N(\mathcal{G})$ is a Kan complex. Given any horn $\Lambda^n_k \to N(\mathcal{G})$, the missing face can be filled using compositions and inverses of the given morphisms.

For instance, a $\Lambda^2_0$-horn in $N(\mathcal{G})$ gives morphisms $g: c_1 \to c_2$ and $h: c_0 \to c_2$. The filler requires $f: c_0 \to c_1$ with $g \circ f = h$, which is $f = g^{-1} \circ h$.

A $\Lambda^2_2$-horn gives $f: c_0 \to c_1$ and $h: c_0 \to c_2$. The filler requires $g: c_1 \to c_2$ with $g \circ f = h$, so $g = h \circ f^{-1}$.

For a group $G$, the nerve $N(BG)$ is a Kan complex (since $BG$ is a groupoid). Its homotopy groups are:

$\pi_0(N(BG)) = \{*\}, \quad \pi_1(N(BG)) = G, \quad \pi_n(N(BG)) = 0 \text{ for } n \geq 2.$

So $N(BG)$ is a $K(G, 1)$, an Eilenberg--MacLane space. This matches the topological classifying space $BG$.

For $\operatorname{Sing}(S^n)$ (the singular set of the $n$-sphere):

$\pi_k(\operatorname{Sing}(S^n)) = \pi_k(S^n)$

since $\operatorname{Sing}$ preserves homotopy groups. So $\pi_n(\operatorname{Sing}(S^n)) = \mathbb{Z}$, $\pi_3(\operatorname{Sing}(S^2)) = \mathbb{Z}$ (Hopf), etc.

The simplicial homotopy groups agree with the topological ones via geometric realization.

A Kan complex $X$ is contractible if $\pi_0(X) = \{*\}$ and $\pi_n(X, x) = 0$ for all $n \geq 1$ and all basepoints $x$. Equivalently, the map $X \to \Delta[0]$ is a weak equivalence.

For example, $\operatorname{Sing}(\mathbb{R}^n)$ is contractible for all $n$, as is $\operatorname{Sing}(|\Delta^n|)$.

A map $p: X \to Y$ is a trivial Kan fibration (also called an acyclic fibration) if it has the right lifting property with respect to all boundary inclusions $\partial\Delta[n] \hookrightarrow \Delta[n]$.

Trivial Kan fibrations are both Kan fibrations and weak equivalences. If $p: X \to \Delta[0]$ is a trivial Kan fibration, then $X$ is contractible.

Any simplicial set $X$ admits a trivial Kan fibration $\widetilde{X} \to X$ where $\widetilde{X}$ is a contractible Kan complex. This is the simplicial analogue of a path space.

If $p: X \to Y$ is a Kan fibration and $y \in Y_0$ is a vertex, the fiber $p^{-1}(y) = X \times_Y \{y\}$ is a Kan complex. The long exact sequence of homotopy groups applies:

$\cdots \to \pi_n(p^{-1}(y)) \to \pi_n(X) \to \pi_n(Y) \to \pi_{n-1}(p^{-1}(y)) \to \cdots$

This is the simplicial analogue of the long exact sequence for a fibration in topology.

For a Kan complex $X$ with basepoint $x \in X_0$, the path space $PX$ is defined so that $(PX)_n$ consists of $(n+1)$-simplices $\sigma$ of $X$ with $d_0(\sigma) = s_0^n(x)$ (the degenerate $n$-simplex at $x$). The evaluation map $PX \to X$ sending $\sigma$ to $d_1 d_2 \cdots d_{n+1}(\sigma)$ is a Kan fibration, with fiber the loop space $\Omega X$.

This gives $\pi_n(X) \cong \pi_{n-1}(\Omega X)$, the fundamental looping-delooping relationship.

Every Kan complex $X$ contains a minimal sub-Kan complex $X_{\min} \subseteq X$ that is a deformation retract. The inclusion $X_{\min} \hookrightarrow X$ is a weak equivalence.

Moreover, two minimal Kan complexes are weakly equivalent if and only if they are isomorphic. This provides a canonical "normal form" for each homotopy type.

For example, the minimal Kan complex representing $K(\mathbb{Z}, 1)$ has one $0$-simplex, $\mathbb{Z}$ many non-degenerate $1$-simplices, and its higher simplices are determined by the group structure.