The $\omega$-nerve functor
from ∞-categories to simplicial sets is the functor induced by the general logic of nerve and realization from the orientals: the cosimplicial $\omega$-category
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 $C$:
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.
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.
$\omega$-nerve
Last revised on March 11, 2012 at 19:30:38. See the history of this page for a list of all contributions to it.