nLab necklace

Definition

A necklace is a gluing of simplicial sets

Δ n 1Δ n 2Δ n k.\Delta^{n_1}\vee \Delta^{n_2}\vee\cdots\vee \Delta^{n_k}.

Here the final vertex of Δ n i\Delta^{n_i} is identified with the initial vertex of Δ n i+1\Delta^{n_{i+1}}.

Applications

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 XX, construct a simplicial category by taking its objects to be vertices of XX, and for a pair of object xx, yy take as the simplicial set of morphisms xyx\to y the simplicial set whose kk-simplices are necklaces of length kk in XX as described above, with the initial vertex of Δ n 1\Delta^{n_1} mapping to xx and the final vertex of Δ n k\Delta^{n_k} mapping to yy.

Dugger and Spivak prove that the resulting simplicial category is weakly equivalent to the other simplicial categories that one can extract from XX, e.g., the left adjoint of the homotopy coherent nerve.

References

A similar idea is developed for complete Segal spaces in

Last revised on May 4, 2024 at 19:52:38. See the history of this page for a list of all contributions to it.