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 $𝟘$, $𝟙$, $𝟚:=𝟙\star𝟘$, $𝟛:=𝟚\star𝟘$ $\ldots$. 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 I.** A more conceptual way to build $\Delta_3$ and its higher dimensional cousins is by using lax joins:
- $\Delta$ is the full subcategory of $\mathsf{Cat}$ spanned by $𝟘$, $𝟙$, and the iterated joins $𝟚:=𝟙\star𝟘$, $𝟛𝟙\star𝟘\star𝟘$,
- $\Delta_2$ is the full subcategory of $\mathsf{2Cat}$ spanned by $𝟘$, $𝟙$, and the iterated *lax* joins $𝟚_2:=𝟙\star^{\mathrm{lax}}𝟘$, $𝟛_2:=𝟙\star^{\mathrm{lax}}𝟘\star^{\mathrm{lax}}𝟘$.

There are also oplax and pseudo versions of this construction, so we should really be speaking of a "lax simplex $3$-category". More generally, there will be $3^{n-1}$ variants of "the simplex $n$-category", just like there are $3^{n-1}$ variants of the join of $n$-categories.¹ Horrifying!

**Aside II.** 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.)

**Question 2 stated, finally.** 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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¹There's a similar phenomenon in the cubical word:
- The "cube $2$-category" may be defined as the full subcategory of $\mathsf{Cat}$ spanned by $𝟘$, $𝟙$, $𝟙\times𝟙$, $𝟙\times𝟙\times𝟙$, $\ldots$: instead of iterated joins, this time we use iterated products.
- The "cube $3$-category" is then defined as the full subcategory of $\mathsf{2Cat}$ spanned by $𝟘$, $𝟙$, $𝟙\otimes_{\mathrm{Gray}}𝟙$, $𝟙\otimes_{\mathrm{Gray}}𝟙\otimes_{\mathrm{Gray}}𝟙$, $\ldots$, where this time we throw away the Cartesian product and use the Gray tensor product.

As with the simplicial case, there will be $3^{n-1}$ variants of "the cube $n$-category", just like there are $3^{n-1}$ variants of the Gray tensor product of $n$-categories. Also horrifying!