---
tag: 4BW0Y31347
kind: definition
name: Horn
msc: [55U10, 18N50]
defines: D5Z60BRP18
sources:
  - url: https://en.wikipedia.org/wiki/Simplicial_set
---

## Definition

For $n \ge 0$ let $[n] = \set{0, 1, \dots, n}$, and let $\Delta^n$ be the standard $n$-simplex simplicial set, whose $m$-simplices are the order-preserving maps $[m] \to [n]$.
Let $n \ge 1$ and $0 \le k \le n$.
The *horn* $\Lambda^n_k$ is the simplicial subset of $\Delta^n$ whose $m$-simplices are the order-preserving maps $\alpha \colon [m] \to [n]$ such that
$$\alpha([m]) \cup \set{k} \ne [n].$$
The horn is *inner* if $0 < k < n$ and *outer* if $k = 0$ or $k = n$.

## Remarks

Equivalently, $\Lambda^n_k$ is the union of the faces $d^i \Delta^{n-1} \subseteq \Delta^n$ for $i \ne k$, where $d^i$ is the face that omits the vertex $i$.
Its geometric realization is the boundary of the $n$-simplex with the interior of the face opposite the vertex $k$ removed.

Example: $\Lambda^2_1$ consists of the edges $\set{0, 1}$ and $\set{1, 2}$ and their vertices.
The edge $\set{0, 2}$ is the map $[1] \to [2]$ with image $\set{0, 2}$, and $\set{0, 2} \cup \set{1} = [2]$, so it is not in $\Lambda^2_1$.
The edge $\set{0, 1}$ has image $\set{0, 1}$, and $\set{0, 1} \cup \set{1} \ne [2]$, so it is in $\Lambda^2_1$.

Non-example: the boundary $\partial \Delta^2$ contains all three edges, so it is not a horn.
The edge $\set{0, 2}$ fails the condition for $k = 1$, the edge $\set{1, 2}$ for $k = 0$, and the edge $\set{0, 1}$ for $k = 2$.

A map $\Lambda^n_k \to X$ to a simplicial set $X$ is the same as a family of $n$ simplices $x_i \in X_{n-1}$, $i \ne k$, that agree on their common faces.