Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

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