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

## Summary

A horn is the union of all the codimension-one faces of a standard simplex except one.
In the theory of simplicial sets, horns are the shapes against which one tests extension properties.
A Kan complex is a simplicial set in which every horn can be filled.
A quasi-category is one in which every inner horn can be filled.

## Explanations

### L1 human

Take a triangle and keep only two of its three edges.
The two edges form a corner shaped like the letter V, which is the horn $\Lambda^2_k$, where $k$ is the vertex where the two edges meet.
Filling the horn means finding the third edge and the triangle itself.
In a space, a horn of paths can always be filled.
Two paths that meet at a point compose to a third path, and a triangle witnesses the composition.