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$\DeclareMathOperator\Ex{Ex}$Kan's $\Ex^\infty$ functor (see Why is Kan's $Ex^\infty$ functor useful?) $\Ex^\infty\colon\mathsf{sSets}\to\mathsf{Kan}$ produces a Kan complex from a simplicial set in a combinatorial way, providing an $\infty$-groupoidification functor $\Ex^\infty\colon\mathsf{Cats}_{\infty}\to\mathsf{Grpd}_\infty$ in the context of $\infty$-categories (modelled as quasicategories). In addition, it comes with a natural transformation $\mathfrak{c}\colon\operatorname{id}_{\mathsf{sSets}}\Rightarrow\Ex^\infty$ whose components $$\mathfrak{c}_{X}\colon X_{\bullet}\to\Ex^\infty_\bullet(X)$$ are weak homotopy equivalences.

It is then natural to wonder if there exist analogues of $\Ex^\infty$ with Kan complexes (which model $\infty$-groupoids, i.e. $(\infty,0)$-categories) replaced with general $(\infty,n)$-categories; this would be a functor $$Q_\bullet\colon\mathsf{sSets}\to\mathsf{Cats}_{(\infty,n)}$$ producing an $(\infty,n)$-category from a simplicial set, and restricting (in some suitable sense, depending on the chosen model for $(\infty,n)$-categories) to the left adjoint $$\mathsf{triv}_n\colon\mathsf{Cats}_{(\infty,n)}\to\mathsf{Cats}_{(\infty,n-1)}$$ to the inclusion $\iota_n\colon\mathsf{Cats}_{(\infty,n-1)}\hookrightarrow\mathsf{Cats}_{(\infty,n)}$.

For the $n=1$ case (see Dmitri Pavlov's answer to A combinatorial approximation functor sSet->qCat), one can essentially just pick a simplicial set $X_\bullet$ and apply $\mathrm{Ex}^\infty$ to its morphism simplicial sets.

Similarly, is there a combinatorial/non-recursive (in the sense described here) analogue of the pair $(\mathrm{Ex}^\infty,\mathfrak{c})$ for:

  • $(\infty,2)$-categories defined as in Kerodon? (I guess one could perhaps repeat Dmitri's strategy with $\mathrm{Q}_\bullet$ instead of $\Ex^\infty$, at least as long as there are analogues of $\mathfrak{C}\dashv\mathrm{N}^\mathrm{hc}_\bullet$ for categories enriched in quasicategories and $(\infty,2)$-categories, or perhaps by using $2$-fold complete Segal spaces instead of these, or (…).)
  • $(\infty,n)$-categories (including $(\infty,\infty)$-categories) modelled as complicial sets?
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    $\begingroup$ Model categories have functorial fibrant replacements. $\endgroup$
    – Fernando Muro
    Commented Jan 7, 2022 at 20:09
  • $\begingroup$ For complicial sets, you can presumably just forget the markings and then apply the usual $Ex^\infty$ functor $\endgroup$
    – Tim Campion
    Commented Jan 7, 2022 at 20:09
  • $\begingroup$ @FernandoMuro I've edited the question to be more explicit about this: I was instead hoping for a more combinatorial/non-recursive construction (maybe a right adjoint functor or a colimit of right adjoint functors, along the lines of Tim's question here) $\endgroup$
    – Emily
    Commented Jan 7, 2022 at 20:53
  • $\begingroup$ @TimCampion Is there a (combinatorial, in the sense of your question here) construction that instead starts with a simplicial set and then builds a complicial/comical set from it, though? $\endgroup$
    – Emily
    Commented Jan 7, 2022 at 20:57
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    $\begingroup$ Ah, I think I understand the question now. Certainly it's going to depend on your model. I'm tempted to guess that for $n$-fold complete Segal spaces, you could apply $Ex^\infty$ to just one simplicial direction. For complicial sets it seems difficult. (And yes, I meant that $X^\sharp$ is a saturated complicial set.) $\endgroup$
    – Tim Campion
    Commented Jan 11, 2022 at 12:35

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