Spahn nerve

References

General theory of realization-and-nerve

• Tom Leinster, How I learned to love the nerve construction (2008) (blog)

The associated paper is

• Mark Weber, Familial 2-functors and parametric right adjoints (2007) (tac)

These ideas are clarified and expanded on in

• Clemens Berger, Paul-André Melliès, Mark Weber, Monads with Arities and their Associated Theories (2011) (arXiv:1101.3064)

• monad with arities

Particular instances

• Emily Riehl, Understanding the Homotopy Coherent Nerve, blog

• nerve of a graph, at Segal condition

• Complicial sets are precisely those simplicial sets which arise as the omega nerve of a strict omega-category.

• A simplicial set is the nerve of a category precisely if it satisfies the Segal condition, at nerve

• (Nerve Theorem, Segal 1968): A simplicial set is the nerve of a small category precisely if it satsfies the Segal conditions. See the reference at Segal condition

• For further relations between nerves and Segal condition, see nerves and Segal conditions