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# Duoidal category

Last updated May 29, 2023

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## # 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.

• $(ψ_2 ⊗ id) ⨾ ψ_2 ⨾ (α ◁ α) = α ⨾ (id ⊗ ψ_2) ⨾ ψ_2$, for the associator;
• $(ψ_0 ⊗ id) ⨾ ψ_2 ⨾ (λ ◁ λ) = λ$, for the left unitor; and
• $(id ⊗ ψ_0) ⨾ ψ_2 ⨾ (ρ ◁ ρ) = ρ$, for the right unitor;
• $α ⨾ (id ⊗ φ_2) ⨾ φ_2 = (φ_2 ⊗ id) ⨾ φ_2$, for the associator;
• $(φ_0 ⊗ id) ⨾ φ_2 = λ$, for the left unitor; and
• $(id ⊗ φ_0) ⨾ φ_2 = ρ$, for the right unitor.

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

• $(β ⊗ β) ⨾ ψ_2 ⨾ (id ◁ ψ_2) = ψ_2 ⨾ (ψ_2 ◁ id) ⨾ β$, between associator and tensor;
• $ψ_0 ⨾ (I ◁ ψ_0) = ψ_0 ⨾ (ψ_0 ◁ id) ⨾ β$, between associato`r and unit; (…)

## # 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.

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