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.

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.)

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.

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 $𝟘$, $𝟙$, $𝟚$, $\ldots$. 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:

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?

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!

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:

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?

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!

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 $𝟘$, $𝟙$, $𝟚$, $\ldots$. 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 same 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.

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

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 $𝟘$, $𝟙$, $𝟙\otimes^{\mathrm{lax}}_{\mathrm{Gray}}𝟙$, $𝟙\otimes^{\mathrm{lax}}_{\mathrm{Gray}}𝟙\otimes^{\mathrm{lax}}_{\mathrm{Gray}}𝟙$, $\ldots$.

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 $𝟘$, $𝟙$, $𝟚$, $\ldots$. 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, which are pictured in detail here.

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?

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!