Related:
- duoidal category
- Duoidal Tambara
- Promonoidal category of optics
- Monoidal spliced arrows
- promonoidal category
- Produoidal contour
- produoidal normalization
References:
- Tannaka Duality and Convolution for Duoidal Categories (Booker, Street, 2013)
- The Produoidal Algebra of Process Decomposition (Earnshaw, Hefford, Román, 2023)
Intuition
Produoidal categories are the profunctorial counterpart of duoidal categories. Every produoidal category can be normalized to a normal produoidal. The cofree produoidal category over a category is the monoidal spliced arrow category, and its normalization is the category of monoidal contexts. The notion of representable homomorphism between produoidal categories is that of produoidal functor.
Produoidal categories as virtual duoidal categories
Produoidal categories can be seen as virtual duoidal categories where morphisms factor. In this case, the two different directions of laxity conflict, and we may want to consider them as opvirtual duoidal categories.
Normal produoidal categories
A normal produoidal category is a produoidal category where .