nLab
omega-nerve

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higher category theory

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Definition

The ω\omega-nerve functor

N:ωCatsSet N : \omega Cat \to sSet
CN(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 ω\omega-category

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

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

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

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

Characterization of ω\omega-categories by their ω\omega-nerves

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

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

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

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

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

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

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