# nLab produoidal category

Produoidal category

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

# 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

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