Applications of the simplex $2$-category and its higher dimensional cousins

Question

The 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; (see also the nCatCafé discussion here)
• On cofinal functors of $\infty$-bicategories.

(Also worth of note is the following fact, noted 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 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, pictured in detail here.

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 $𝟚:=𝟙\star𝟘$, $𝟛:=𝟙\star𝟘\star𝟘$,
• $\Delta_2$ 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 close to (but not quite, I think) the lax morphism classifier $\mathsf{L}\Delta_2$ of the simplex $2$-category, for which we have a bijection $$\{\text{$2$-functors $\mathsf{L}\Delta_2\to\mathcal{C}$}\}\cong\{\text{Lax functors $\Delta_{2}\to\mathcal{C}$}\}.$$

Aside III. There are actually lax, oplax, and pseudo variants of the simplex $1$-category too: the lax variant is the usual simplex category, the oplax variant is isomorphic to the lax variant, and the pseudo variant is the full subcategory of $\mathsf{Cat}$ spanned by the localisations of the ordinal categories at every morphism.

Alternatively, we may also construct the pseudo variant as the full subcategory of $\mathsf{Cat}$ spanned by the "iso-ordinal" categories, defined iteratively starting from $𝟘$ by means of the "isojoin" of categories, defined in the same way as the join $\mathcal{C}\star\mathcal{D}$ but where we replace the morphisms from the objects of $\mathcal{C}$ to those of $\mathcal{D}$ in $\mathcal{C}\star\mathcal{D}$ by isomorphisms.

And it turns out that the resulting category is already very well-known: it is a skeleton of the category of finite sets, and presheaves on it are symmetric (simplicial) sets, which form another model for $\infty$-groupoids.

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?

---

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!