Recall that given two strict ω-categories $A$ and $B$, their lax Gray tensor product $A\otimes B$ is sent to the Verity-Gray tensor product of their associated complicial sets $N_{\operatorname{Strat}}(A) \otimes N_{\operatorname{Strat}}(B)$ by the stratified nerve of Verity. Further, given any strict ω-category $A$, the underlying simplicial set of the complicial set $N_{\operatorname{Strat}}(A)$ is exactly $N_\omega(A)$, where $N_\omega = N_{\mathcal{O}}$ is the nerve functor associated with the cosimplicial object $\mathcal{O}:\Delta \to \omega\operatorname{-cat}$ where $\mathcal{O}[n]$ is $n$th oriental as defined by Street.

Consider the following case: If we take the lax Gray tensor product of two freestanding 1-cells $[1]\otimes [1]$ and apply the stratified nerve, we obtain a Verity-Gray tensor product $N_{\operatorname{Strat}}([1])\otimes N_{\operatorname{Strat}}([1])$. By the definition of this tensor product, its underlying simplicial set is given simply as $[1]\times [1]$, and therefore, we see that the Street nerve $N_\omega([1]\otimes [1])=[1]\times [1]$.

If we actually take a moment to draw out the strict ω-category $[1]\otimes [1]$, we see that it can be visualized as:

```
•====•--->•
|\ \ |
| \ \ |
| \ =>\ |
| \ \ |
v v vv
•--->•====•
```

where the "====" means that we are identifying the vertices on either end.

Also, the second oriental $\mathcal{O}[2]$ is traditionally written as:

```
•---->•
\ |
\=> |
\ |
\ |
vv
•
```

but the strict ω-category that this generates can be visualized as:

```
•====•--->•
\ \ |
\ \ |
\ =>\ |
\ \ |
v vv
•====•
```

But $[1]\times [1]$ viewed as a simplicial set is just the union of its two nondegenerate $2$-simplices. These two nondegenerate $2$-simplices should correspond to maps of strict ω-categories $\mathcal{O}[2] \to [1]\otimes [1]$. The bottom-left $2$-simplex is obviously given by the map sending $\mathcal{O}[2]$

```
•===•
|\ \
| \ \
| \<= \
| \ \
v v v
•--->•===•
```

onto the bottom-left simplex

```
•
|\
| \
| \
| \
v v
•--->•
```

by collapsing the 2-cell (note the flipped orientation)

```
•===•
\ \
\ \
\<= \
\ \
v v
•===•
```

to an edge.

The top-right $2$-simplex of $[1]\times [1]$ classifies the inclusion of $\mathcal{O}[2]$

```
•====•--->•
\ \ |
\ \ |
\ =>\ |
\ \ |
v vv
•====•
```

in $[1]\otimes [1]$

```
•====•--->•
|\ \ |
| \ \ |
| \ =>\ |
| \ \ |
v v vv
•--->•====• .
```

The thing I don't understand is why some of the other maps $\mathcal{O}[2]\to [1]\otimes [1]$ classify degenerate 2-faces in $[1]\times [1]$.

For instance, consider either of the maps $\mathcal{O}[2] \to D_2$

sending

```
•====•--->•
\ \ |
\ \ |
\ =>\ |
\ \ |
v vv
•====•
```

onto

```
•===•
\ \
\ \
\ =>\
\ \
v v
•===•
```

whose restriction to the subobject

```
•---->•
\ |
\ |
\ |
\ |
vv
•
```

is given by a codegeneracy (collapsing this simplex either to 0 0 2 or 0 2 2).

Then since $D_2$ embeds in $[1]\otimes [1]$, we obtain a map $\mathcal{O}[2] \to [1]\otimes [1]$ that doesn't appear to be degenerate.

However, it follows from the description of $N_\omega([1]\otimes [1])=[1]\times [1]$ that these maps must classify degenerate 2-simplices. Why are the simplices classified by these maps degenerate?