# Duoidal category

Duoidal categories sidestep the Eckmann-Hilton argument using an extra dimension to lax the interaction between two monoids: they are pseudomonoids in the 2-category of monoidal categories. *Duoidal categories* can be used for
signalling in process theories and to refine
braided monoidal categories.

Duoidal categories have multiple interesting variants: normal duoidal categories identify both units; physical duoidal categories ask for enough symmetry to talk about dependencies; virtual duoidal categories provide the algebra of shuffling; produoidal categories can be used to study decomposition.

A coherence theorem for duoidal categories can be found in the monograph by Aguiar and Mahajan (2009). One needs to be careful, though, because it looks wrong as stated: duoidal categories are not fully coherent – only physical duoidal categories are.

*Related.*

- Strict duoidal category
- Strings for duoidal categories
- compositional dependencies with duoidals
- coherence for duoidal categories
- Duoids, bimonoids in a duoidal
- monoidal multicategory
- produoidal category
- Conjecture - Every duoidal embeds into an adjoint pseudomonoid

Normal and physical duoidal categories.

- normal duoidal category
- duoidal normalization
- physical duoidal category
- mixing normal duoidals
- Two-sided duoidal duality

*References*

- Commutativity (Garner, Lopez Franco, 2015)
- Duoidal Structures for Compositional Dependence (Shapiro, Spivak, 2022)
- String Diagrams for Physical Duoidal Categories (Rajesh, Román, 2024)
- The Produoidal Algebra of Process Decomposition (Earnshaw, Hefford, Román, 2023)
- Monoidal Context Theory, PhD Thesis (Román, 2023)