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The Joyal rigidification functor is defined as the left adjoint of the Cordier–Vogt homotopy coherent nerve functor.
The Dugger–Spivak rigidification functor provides a more explicit model for the same (∞,1)-functor, by virtue of writing down an explicit model that does not use colimits.
Specifically, given a simplicial set $S$ (not necessarily fibrant in the Joyal model structure), we construct the Dugger–Spivak rigidification $\mathfrak{C}^{\mathrm{nec}}S$ as the following simplicial category. Objects are vertices of $S$.
The simplicial set of morphisms $x\to y$ is the nerve of the category of necklaces in $S$ (introduced by Hans-Joachim Baues). A necklace is a simplicial map
$\Delta^{n_1}\vee \cdots \vee \Delta^{n_k}\to S,$where $\vee$ glues the final vertex of the preceding simplex to the initial vertex of the following simplex. Morphisms are commutative triangles of simplicial maps that preserve the initial and final vertex of the entire necklace.
Composition is defined by concatenating necklaces. The resulting functor from simplicial sets admits a zigzag of weak equivalences to the Joyal rigidification functor.
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