DanielFilan comments on Multiplicative Operations on Cartesian Frames

[DanielFilan]

This thread is mostly me trying to work something out and reporting the results. To the extent there’s a question, it’s this: if C ◃ D, is there something interesting to say about C ⊸ D?

[Scott Garrabrant]

I am not sure, but I think that the answer is that you can’t say anything interesting with just $◃$, but can maybe say interesting things with ${◃}_{+}$ and ${◃}_{\times }$, which I am about to introduce. In the post that just went up, ${◃}_{+}$ is the relationship between one of the components and a sub-sum, and ${◃}_{\times }$ is the relationship between one of the components and a sub-tensor.$◃$ is the transitive closure of ${◃}_{+}$ and ${◃}_{\times }$.

I think that if $C{◃}_{+}D$, then there is a nice morphism from $C$ to $D$, and if $C{◃}_{\times }D$, there is a set of nice morphisms from $C$ to $D,$ but in some degenerate cases that set is empty, which is how I constructed a counter example in my other comment.