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A necklace is a gluing of simplicial sets
Δn1∨Δn2∨⋯∨Δnk.Here the final vertex of Δni is identified with the initial vertex of Δni+1.
Necklaces provide a way to extract a simplicial category from a simplicial set (not necessarily fibrant) in the Joyal model structure.
Given such a simplicial set X, construct a simplicial category by taking its objects to be vertices of X, and for a pair of object x, y take as the simplicial set of morphisms x→y the simplicial set whose k-simplices are necklaces of length k in X as described above, with the initial vertex of Δn1 mapping to x and the final vertex of Δnk mapping to y.
Dugger and Spivak prove that the resulting simplicial category is weakly equivalent to the other simplicial categories that one can extract from X, e.g., the left adjoint of the homotopy coherent nerve.
Daniel Dugger, David I. Spivak, Rigidification of quasi-categories, arXiv:0910.0814.
Daniel Dugger, David I. Spivak, Mapping spaces in Quasi-categories, arXiv:0911.0469.
A similar idea is developed for complete Segal spaces in
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