$\DeclareMathOperator\Ex{Ex}$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) replaced with general $(\infty,n)$-categories. 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?

$\DeclareMathOperator\Ex{Ex}$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?

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

It is then natural to wonder if there exist analogues of $\mathrm{Ex}^\infty$ with Kan complexes (which model $\infty$-groupoids) replaced with general $(\infty,n)$-categories. For the $n=1$ case, 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 $\mathrm{Ex}^\infty$, 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?

$\DeclareMathOperator\Ex{Ex}$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) replaced with general $(\infty,n)$-categories. 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?

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

It is then natural to wonder if there exist analogues of $\mathrm{Ex}^\infty$ with Kan complexes (which model $\infty$-groupoids) replaced with general $(\infty,n)$-categories. For the $n=1$ case (see here and here), one can essentially just pick a simplicial set $X_\bullet$ and apply $\mathrm{Ex}^\infty$ to the morphism spaces of $X_\bullet$:

1. First we construct a simplicial category $\mathfrak{C}(X)$ from $X_\bullet$;
2. Then we apply the change of enrichment base functor $\mathsf{Cats}_{\mathsf{sSets}}\to\mathsf{Cats}_{\mathsf{Kan}}$ associated to $\mathrm{Ex}^\infty$ (which is monoidal);
3. And finally we apply the homotopy coherent nerve to the result, obtaining a quasicategory $\mathrm{Q}_\bullet(X)$

Analogously to the case for Kan complexes, we have a natural transformation $\mathfrak{c}\colon\mathrm{id}_{\mathsf{sSets}}\Rightarrow\mathrm{Q}$, whose components $$\mathfrak{c}_{X}\colon X_{\bullet}\to\mathrm{Q}_\bullet(X)$$ are categorical equivalences.

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 $\mathrm{Ex}^\infty$, 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?

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

It is then natural to wonder if there exist analogues of $\mathrm{Ex}^\infty$ with Kan complexes (which model $\infty$-groupoids) replaced with general $(\infty,n)$-categories. For the $n=1$ case, 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 $\mathrm{Ex}^\infty$, 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?