The [simplex category](https://ncatlab.org/nlab/show/simplex+category) $\Delta$ has a $2$-category refinement $\Delta_2$ given by the full sub-$2$-category of the $2$-category $\mathsf{Cat}$ spanned by the ordinal categories $𝟘$, $𝟙$, $𝟚$, $\ldots$. One fun fact about it is that the coface and codegeneracy maps in this $2$-category now form an adjoint string:
$$\mathrm{d}_0\vdash\mathrm{s}_0\vdash\mathrm{d}_1 \vdash \dots \vdash\mathrm{d}_{n-1}\vdash \mathrm{s}_{n-1} \vdash\mathrm{d}_n.$$
I've seen this extra structure only used two times, namely:
- [A categorified Dold-Kan correspondence](https://arxiv.org/abs/1710.08356); (see also the nCatCafé discussion [here](https://golem.ph.utexas.edu/category/2017/10/categorification_and_the_cosmi.html))
- [On cofinal functors of $\infty$-bicategories](https://d-nb.info/1264307314/34).

(Also worth of note is the following fact, [noted](https://golem.ph.utexas.edu/category/2017/10/categorification_and_the_cosmi.html#c052895) in the nCatCafé discussion linked above: If the simplicial nerve of a $2$-monad $T$ extends to a $2$-simplicial nerve, then $T$ is lax-idempotent.)

**Question 1.** Other than these, what are some other applications/structures which use the $2$-category structure of $\Delta_2$?

**Question 2.** It seems to me that $\Delta_2$ is built by means of the following procedure, specialised to $k=2$:
- First we take Street's [orientals](https://ncatlab.org/nlab/show/oriental) and ask that every $n$-cell with $n\geq k$ is an identity;
- Then we consider the full subcategory of $(k-1)\text{-}\mathsf{Cat}$ spanned by the $(k-1)$-categories corresponding to these "truncated orientals".

Repeating this procedure for $n\geq3$ should give a simplex $3$-category $\Delta_3$, a simplex $4$-category $\Delta_4$, and so on. For example, the $2$-categories $𝟘_2$, $𝟙_2$, $𝟚_2$, $\ldots$ used to build $\Delta_3$ look just like the $n$-simplices in the Duskin nerve of a $2$-category, which are pictured in detail [here](https://ncatlab.org/nlab/show/geometric+nerve+of+a+bicategory#picturing_the_duskin_nerve). 

*Aside.* Being a strict $n$-category, $\Delta_n$ also has an underlying $(n-1)$-category $\Delta_{n|1}$ (e.g. $\Delta_{2|1}$ is the usual simplex category). It seems to me that $\Delta_{3|1}$ is something like the lax morphism classifier of the simplex $2$-category, so that we have a bijection
$$\{\text{$2$-functors $\Delta_{3|1}\to\mathcal{C}$}\}\cong\{\text{Lax functors $\Delta_{2}\to\mathcal{C}$}\},$$
(although I haven't fully checked this.)

Have these simplex $n$-categories been considered before in the literature (or by someone reading this post)? What would be some applications of them?

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**Edit:** I just noticed that for question 2 there are also "pseudo", and "oplax" versions of the simplex $n$-category for $n\geq3$: we could take the appropriate $n$-morphisms in the orientals to be isomorphisms, or to go in the opposite direction.

In the cubical setting, this seems to be related to the lax, pseudo, and oplax variants of the Gray tensor product:
- We could define the cube $2$-category $\square_2$ as the full subcategory of the $2$-category $\mathsf{Cat}$ spanned by the categories $𝟘$, $𝟙$, $𝟙\times𝟙$, $𝟙\times𝟙\times𝟙$, $\ldots$.
- For the cube $3$-category, we can pick any of the three Gray tensor products $\otimes^\mathrm{lax}_{\mathrm{Gray}}$, $\otimes^\mathrm{oplax}_{\mathrm{Gray}}$, $\otimes^\mathrm{ps}_{\mathrm{Gray}}$.
 
  So e.g. the "lax cube $3$-category" $\square^\mathrm{lax}_3$ would be the full subcategory of the $3$-category of $2$-categories spanned by the $2$-categories $𝟘$, $𝟙$, $𝟙\otimes^{\mathrm{lax}}_{\mathrm{Gray}}𝟙$, $𝟙\otimes^{\mathrm{lax}}_{\mathrm{Gray}}𝟙\otimes^{\mathrm{lax}}_{\mathrm{Gray}}𝟙$, $\ldots$.

**Edit 2:** Even worse, there's such a choice of orientation for each non-invertible $n$-cell with $n\geq2$, so that there are $3^{n-1}$ variants of "the simplex $n$-category", just like there are $3^{n-1}$ variants of the Gray tensor product of $n$-categories. Horrifying!