5
$\begingroup$

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 structure on simplicial sets. Is this in fact the case? If so, has this duoidal category been studied before?

Improve this question
$\endgroup$

1 Answer 1

Reset to default
4
$\begingroup$

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.

Improve this answer
$\endgroup$
1
  • 2
    $\begingroup$ Perhaps more interestingly, it looks like the join actually distributes over the product (yes, not the other way around!) but only colaxly. But on reflection, I think this too is automatic for monoidal categories with finite products. $\endgroup$
    – Tim Campion
    Commented May 9, 2018 at 4:37

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .