nLab produoidal category

Idea

A produoidal category is like a duoidal category in whose structure (namely, the two tensor products and unit objects) we have replaced functors by profunctors. Alternatively, a produoidal category is a category with two promonoidal structures which interchange laxly.

Definition

A produoidal category is a pair of pseudomonoids that interchange laxly in the monoidal bicategory Prof. This means that it is a category $\mathbb{C}$ together with

• a first promonoidal structure $\mathbb{C}_{\otimes} \colon \mathbb{C} \times \mathbb{C} ⇸ \mathbb{C}$ and $\mathbb{C}_{I}\colon 1 ⇸ \mathbb{C}$;
• a second promonoidal structure $\mathbb{C}_{\triangleleft} \colon \mathbb{C} \times \mathbb{C} ⇸ \mathbb{C}$ and $\mathbb{C}_{N}\colon 1 ⇸ \mathbb{C}$;
• the same laxators as a duoidal category, including, for instance,
  $$\begin{aligned} & \int^{U,V}\mathbb{C}_{\otimes}(X;U,V) \times \mathbb{C}_{\triangleleft}(U;A,B) \times \mathbb{C}_{\triangleleft}(V;C,D) \to \int^{P,Q}\mathbb{C}_{\triangleleft}(X;P,Q) \times \mathbb{C}_{\otimes}(P;A,C) \times \mathbb{C}_{\otimes}(Q;B,D); \end{aligned}$$
• the same coherence conditions as a duoidal category.

Examples

• Optics over a monoidal category form a produoidal category. EHR23

References

The first mention of produoidal categories as a duoidale seems to be:

• Thomas Booker?, Ross Street. Tannaka duality and convolution for duoidal categories, (link).

An explicit unpacking of the definition, along with examples including the category of optics appears in

• Matt Earnshaw, James Hefford?, Mario Román. The Produoidal Algebra of Process Decomposition (link).