nLab necklace

Definition

A necklace is a gluing of simplicial sets

$$\Delta^{n_1}\vee \Delta^{n_2}\vee\cdots\vee \Delta^{n_k}.$$

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

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 $X$, 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 $x\to y$ the simplicial set whose $k$-simplices are necklaces of length $k$ in $X$ as described above, with the initial vertex of $\Delta^{n_1}$ mapping to $x$ and the final vertex of $\Delta^{n_k}$ 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.

Related concepts

• homotopy coherent nerve

• simplicial category, quasicategory

References

• 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

• Shaul Barkan, Jan Steinebrunner, Segalification and the Boardman-Vogt tensor product, arXiv:2301.08650.