Return to Answer

This is the subject of an unpublished manuscript of Ross Street.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 $\mathbf{2\text{-Cat}}$.

This is the subject of an unpublished manuscript of Ross Street.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 $\mathbf{2\text{-Cat}}$.)

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.

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 $\mathbf{2\text{-Cat}}$.

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 structure on $\mathbf{\text{Cu}}$, one could also witness in this manner the pseudo Gray tensor product as a left 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.