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The associated paper is
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)
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