---
tag: 779ZSEF36C
kind: definition
name: Nerve of a category
msc: [18N50, 55U10]
defines: RV0MKJ3CQ6
uses: [A2DE66937D, 28NQ0KRHD3, FRFYHJGAN3, WW81ARB8K9]
sources:
  - url: https://en.wikipedia.org/wiki/Nerve_(category_theory)
---

## Definition

Let $\mathcal{C}$ be a small [[category|A2DE66937D]], and let $\Delta$ be the [[simplex category|FRFYHJGAN3]].
View each object $[n]$ of $\Delta$ as the category with objects $0, 1, \dots, n$ and exactly one morphism $i \to j$ when $i \le j$.
The *nerve* of $\mathcal{C}$ is the [[simplicial set|WW81ARB8K9]] $N\mathcal{C}$ with
$$N\mathcal{C}_n = \set{\text{functors } [n] \to \mathcal{C}},$$
where an order-preserving map $\theta \colon [m] \to [n]$ acts by precomposition, sending $F \colon [n] \to \mathcal{C}$ to $F \circ \theta$.

## Remarks

A [[functor|28NQ0KRHD3]] $[n] \to \mathcal{C}$ is the same as a string $c_0 \xrightarrow{f_1} c_1 \to \cdots \xrightarrow{f_n} c_n$ of composable morphisms.
So $N\mathcal{C}_0$ is the set of objects and $N\mathcal{C}_1$ the set of morphisms.
The face map $d_i$ for $0 < i < n$ composes $f_{i+1} \circ f_i$, while $d_0$ drops $f_1$ and $d_n$ drops $f_n$; the degeneracy $s_i$ inserts an identity at $c_i$.
Example: if $\mathcal{C}$ has one object and only the identity morphism, then $N\mathcal{C}_n$ has one element for each $n$, and $N\mathcal{C}$ is the point $\Delta^0$.
Non-example: the boundary $\partial \Delta^2$ consists of three edges forming a triangle, with no 2-simplex.
It is not the nerve of any category, because in a nerve each pair of composable edges is filled by a 2-simplex.
A simplicial set is isomorphic to a nerve exactly when every inner horn $\Lambda^n_k \to X$ with $0 < k < n$ has a unique filler.