Mario Román


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Duoidal category

Last updated May 29, 2023



# Intuition

By the Eckmann-Hilton argument, each time we have two monoids $(\ast,\circ)$ such that one is a monoid homomorphism over the other, $(a ∘ b) \ast (c ∘ d) = (a \ast c) ∘ (b \ast d)$, we know that both monoids coincide into a single commutative monoid.

However, an extra dimension helps us side-step the Eckmann-Hilton argument. If, instead of equalities or isomorphisms, we use directed morphisms, both monoids (which now may become 2-monoids) do not necessarily coincide, and the resulting structure is that of a duoidal category.

# Definition

A duoidal category is a monoidal category $(ℂ,⊗,I,α,λ,ρ)$ endowed with two lax monoidal functors, the duoidal tensor $(◁)$ and the duoidal unit $(N)$, that provide it with a second monoidal structure. In other words, it is endowed with a duoidal tensor, $(◁) \colon ℂ × ℂ → ℂ$, together with laxators $$ψ_2﹕ (X ◁ Z) ⊗ (Y ◁ W) → (X ⊗ Y) ◁ (Z ⊗ W),$$ $$ψ_0﹕ I → I ◁ I,$$ and a duoidal unit, $N﹕ 1 → ℂ$, together with laxators $$φ_2﹕ N ⊗ N → N,$$ $$φ_0 ﹕ I → N,$$ together with oplax monoidal coherence transformations satisfying the pentagon and the triangle equations.

# Remarks

The duoidal tensor and unit are lax monoidal functors, which means that the laxators must satisfy the following equations.

The coherence transformations $(β,κ,ν)$ are lax monoidal transformations, which means they must satisfy the following equations.

# Notes


# Strict duoidal categories

We say that a duoidal category is strict whenever its base monoidal category is strict and, moreover, the duoidal coherence maps are identities.