The join $\ast$ and the product $\times$ are both important monoidal structures on simplicial sets, but the way they interact is not so simple. For instance, neither distributes over the other. However, I believe there is a comparison map $(A\times B) \ast (C \times D) \to (A \ast C) \times (B \ast D)$, suggesting that they fit together into a [duoidal category](https://ncatlab.org/nlab/show/duoidal+category) structure on simplicial sets. Is this in fact the case? If so, has this duoidal category been studied before?



Do simplicial join and product form a duoidal category structure?
I probably should have read more closely. If $C$ is any monoidal category which is also cartesian monoidal, then the combination of the two monoidal structures is duoidal. This is on the nlab page.


