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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeFeb 24th 2023
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexCreated: ## 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]]. ## References * [[Daniel Dugger]], [[David I. Spivak]], _Rigidification of quasi-categories_, [arXiv:0910.0814](https://arxiv.org/abs/0910.0814). * [[Daniel Dugger]], [[David I. Spivak]], _Mapping spaces in Quasi-categories_, [arXiv:0911.0469](https://arxiv.org/abs/0911.0469). A similar idea is developed for [[complete Segal spaces]] in * Shaul Barkan, Jan Steinebrunner, _Segalification and the Boardmann-Vogt tensor product_, [arXiv:2301.08650](https://arxiv.org/abs/2301.08650). <a href="https://ncatlab.org/nlab/revision/necklace/1">v1</a>, <a href="https://ncatlab.org/nlab/show/necklace">current</a>

   Created:

   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.

   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 Boardmann-Vogt tensor product, arXiv:2301.08650.

   v1, current