# nLab omega-nerve

### Context

#### Higher category theory

higher category theory

# Contents

## Definition

The $\omega$-nerve functor

$N : \omega Cat \to sSet$
$C \mapsto N(C)$

from ω-categories to simplicial sets is the functor induced by the general logic of nerve and realization from the orientals: the cosimplicial $\omega$-category

$O : \Delta \to \omega Cat \,.$

So for $C$ an $\omega$-category, its $\omega$-nerve is the simplicial set whose $k$-simplices are precisely all possible images of the $k$-oriental in $C$:

$N(C)_k := \omega Cat(O[k], C) \,.$

Where the $\omega$-category itself provided rules for how exactly to compose k-morphisms, its $\omega$-nerve just records all possible ways of how $(k+1)$-morphisms connect pasting diagrams of $k$-morphisms in $C$. This is however precisely the same information.

## Characterization of $\omega$-categories by their $\omega$-nerves

Accordingly, omega-nerves may be used to define and identify $\omega$-categories. For instance

• the $\omega$-nerve $N(C)$ is a simplicial set in which all horns have unique fillers precisely if $C$ is a 1-groupoid;

• the $\omega$-nerve $N(C)$ is a simplicial set in which all inner horns have unique fillers precisely if $C$ is an ordinary category;

• the $\omega$-nerve $N(C)$ is a simplicial set in which all horns have any fillers precisely if $C$ is an ∞-groupoid (see Kan complex for more on this);

• the $\omega$-nerve $N(C)$ is a simplicial set in which all inner horns have any fillers precisely if $C$ is an (∞,1)-category.

• in full generality, a simplicial set is the $\omega$-nerve of an $\omega$-category if it is a weak complicial set.

Revised on March 11, 2012 19:30:38 by Tim Porter (95.147.237.7)