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Stream: theory: category theory

Topic: Is Street's nerve fully faithful?

Daniel Teixeira (Oct 18 2025 at 12:57):

In The algebra of oriented simplexes, Street defines a nerve for strict $\omega$-categories via the functor $\mathbf{\Delta}\to\omega\mathsf{Cat}$ given by $[n]\mapsto \mathcal O_n$, the $n$-th oriental. This functor is a left adjoint by an usual Kan extension construction.

Street conjectures that $\omega$-categories are a reflective localization of $\mathsf{sSet}$ via his nerve, i.e. that his nerve if fully faithful. The nLab says that it is faithful. Is it full? Anything I can cite on this?

Daniel Teixeira (Oct 18 2025 at 14:30):

I think the answer is 'no'. In any case, crossposted to the mathoverlfow: https://mathoverflow.net/questions/501761/is-streets-nerve-fully-faithful