---
tag: G87CWTXAWC
kind: definition
name: Quasi-category
msc: [18N60, 18N50]
defines: 9328WEVJEK
uses: [WW81ARB8K9, RV0MKJ3CQ6, FDFYCM3YYC]
sources:
  - url: https://en.wikipedia.org/wiki/Quasi-category
---

## Definition

For $n \ge 0$ let $\Delta^n$ be the standard $n$-simplex, as a [[simplicial set|WW81ARB8K9]].
For $0 \le k \le n$ let $\Lambda^n_k \subset \Delta^n$ be the $k$-th horn: the simplicial subset generated by all faces of $\Delta^n$ except the face opposite the vertex $k$.
The horn is *inner* if $0 < k < n$.
A *quasi-category* is a simplicial set $X$ with the following property.
For every $n \ge 2$, every $0 < k < n$, and every map $\sigma \colon \Lambda^n_k \to X$, there is a map $\tau \colon \Delta^n \to X$ whose restriction to $\Lambda^n_k$ is $\sigma$.

## Remarks

The vertices of $X$ are its objects and the $1$-simplices its morphisms.
For $n = 2$, $k = 1$, the condition says that two morphisms $f \colon x \to y$ and $g \colon y \to z$ have a composite: a $2$-simplex with edges $f$, $g$ and some $h \colon x \to z$.
Example: the [[nerve|RV0MKJ3CQ6]] $N(\mathcal{C})$ of a category $\mathcal{C}$ is a quasi-category, and in it every inner horn has exactly one filler.
Example: every [[Kan complex|FDFYCM3YYC]] is a quasi-category, since it has fillers for all horns, inner or not; the singular simplicial set of a topological space is one.
Non-example: the horn $\Lambda^2_1$ itself is not a quasi-category.
Its identity map $\Lambda^2_1 \to \Lambda^2_1$ does not extend to $\Delta^2$, because every $2$-simplex of $\Lambda^2_1$ is degenerate.
A degenerate $2$-simplex cannot have the two distinct nondegenerate edges $0 \to 1$ and $1 \to 2$ as faces.
Quasi-categories are also called weak Kan complexes or $\infty$-categories.