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Stream: learning: questions

Topic: Heterogeneous duoidal structure

Asad Saeeduddin (Apr 24 2020 at 23:24):

In a duoidal category, there are two monoidal structures $\star, \diamond : \mathcal{C} \times \mathcal{C} \to \mathcal{C}$, and the $\star$ tensor and its unit are monoidal functors with respect to the $\diamond$ monoidal structure. There is a subtlety to what we mean by this: when we say $\star$ is a monoidal bifunctor, what we really mean is the product category $\mathcal{C} \times \mathcal{C}$ has a monoidal structure induced on it by "pairwise" application of the $\diamond$ tensor to pairs of objects and morphisms. It is this "pairwise-$\diamond$ structure" on the domain of $\star$ that is laxly cohered with the $\diamond$ structure on its codomain.

But this concept of the pairwise monoidal structure induced on the product of two monoidal categories works even with distinct monoidal categories. For example, we might take the product and coproduct structures on a bicartesian category $\mathcal{C}$, and on the product category $\mathcal{C} \times \mathcal{C}$ end up with a tensor $\boxtimes : (\mathcal{C} \times \mathcal{C}) \times (\mathcal{C} \times \mathcal{C}) \to \mathcal{C} \times \mathcal{C}$ whose mapping on objects is: $(A, B) \boxtimes (C, D) \mapsto (A \times B, C + D)$ (and so on).

So in a duoidal category the structures laxly cohered by the $\star$ tensor and its unit are a pairwise "doubling" of the $\diamond$ structure on the domain, and the $\diamond$ structure on the codomain. Is there a name for a generalization of this concept where the input and output monoidal structures of the monoidal tensor are allowed to vary independently?

Asad Saeeduddin (Apr 24 2020 at 23:27):

I guess a trivial answer to the question is "monoidal structure whose tensor and unit happen to be monoidal functors", but hopefully there is something a little bit pithier :)