The Gray tensor product as a Kan extension

This is the subject of an unpublished manuscript of Ross Street. Street applies the technique for left Kan extending a monoidal structure along a dense functor from Brian Day's PhD thesis to construct the lax Gray tensor product of 2-categories. He defines a monoidal structure on the full subcategory $\mathbf{\text{Cu}}$ of $\mathbf{2\text{-Cat}}$ whose objects are the "cubes", and shows that the inclusion $K \colon \mathbf{\text{Cu}} \longrightarrow \mathbf{2\text{-Cat}}$ is dense. The lax Gray tensor product is then the left Kan extension of $K \circ \otimes$ along $K \times K$ as in the following diagram (which commutes up to a natural isomorphism).

$$ \require{AMScd} \begin{CD} \mathbf{\text{Cu}} \times \mathbf{\text{Cu}} @>K\times K>> \mathbf{2\text{-Cat}}\times\mathbf{2\text{-Cat}}\\ @V\otimes VV @VV\otimes V\\ \mathbf{\text{Cu}} @>>K> \mathbf{2\text{-Cat}} \end{CD}$$

Note that the inclusion $K$ becomes a strong monoidal functor.

No doubt by suitably altering the monoidal category $\mathbf{\text{Cu}}$, one could also witness in this manner the pseudo Gray tensor product as a left Kan extension.*

*(Edit: This last comment is almost certainly false, since the altered category I imagine I had in mind is not dense in $\mathbf{2\text{-Cat}}$.)

The nice answer given by Alexander can be consider as a particular case of the general framework developed in AM16, specifically in its appendix A.

The objects of the parity complex of cubes in $\infty$-$\mathcal{C}at$ can be defined inductively as the tensor products $\Delta_1 \otimes \dots \otimes \Delta_1$ and they are free objects (in the sense of polygraph/computads), see remark A.16. So it is more natural to define at first the general Gray tensor product for $\infty$-$\mathcal{C}at$ and then truncate to get a Gray tensor product in any degree $n$.

Let $\mathcal{C}_{\text{da}}$ be the category of directed augmented complex and $\mathcal{S}t_{\text f}$ its full subcategory of Steiner complexes, see ch. 2 of AM16. There exists a functor $\nu \colon \mathcal{C}_{\text{da}} \to \infty$-$\mathcal{C}at$ which by a theorem of Steiner is fully faithful when restricted to $\mathcal{S}t_{\text f}$, so that we call the objects of the essential (restricted) image Steiner $\infty$-categories (these are all free in the sense of polygraphs). Another result by Steiner ensures that Steiner complexes are closed under tensor product of directed augmented complexes, see proposition A.4. Moreover, the orientals, the cubes and $\Theta$ are Steiner complexes/categories. Theorem A.14 tells us that the Gray tensor product on $\infty$-$\mathcal{C}at$ is precisely the left Kan extension

$$ \require{AMScd} \begin{CD} \Theta \times \Theta @>\nu\times \nu>> \infty\text{-}\mathcal {C}at\times\infty\text{-}\mathcal{C}at\\ @V\otimes VV @VV\otimes_{\text{Gray}} V\\ \mathcal{S}t_f @>>\nu> \infty\text{-}\mathcal{C}at \end{CD}$$

Finally, by lemma A.25 and proposition A.26, we get the 2-categorical Gray tensor product as the following left Kan extension:

$$ \require{AMScd} \begin{CD} \Theta_2 \times \Theta_2 @>\nu\times \nu>> 2\text{-}\mathcal {C}at\times 2\text{-}\mathcal{C}at\\ @V\otimes VV @VV\otimes_{\text{Gray}} V\\ \mathcal{S}t_f @>>\tau^\text{i}_{\leqslant 2}\circ \nu> 2\text{-}\mathcal{C}at \end{CD},$$ where $\tau^\text{i}_{\leqslant 2}$ is left adjoint to the inclusion functor $2\text{-}\mathcal{C}at \to \infty\text{-}\mathcal{C}at$. The role of $\Theta$ and $\Theta_2$ is that they are small dense sub-categories satisfying a bunch of nice properties, see theorem 6.3. Role which can be played by the 2-trucation of the cubes $\mathbf{Cu}$ for the second case of $2\text{-}\mathcal{C}at$. I don't know whether the full subcategory of cubes is dense in $\infty$-$\mathcal{C}at$, though.