nLab
omega-nerve

Definition

The $\omega$-nerve functor

$$N:\omega \mathrm{Cat}\to \mathrm{sSet}$$N : \omega Cat \to sSet

$$C\mapsto N(C)$$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

$$O:\Delta \to \omega \mathrm{Cat}.$$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 $C$:

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

Related concepts

• nerve

• Duskin nerve

• $\omega$-nerve

• homotopy coherent nerve