# Duoidal category

*Related.*

- Duoid in a duoidal
- Bimonoid in a duoidal
- Strings for duoidal categories
- Compositional dependencies with duoidals
- Produoidal category (+)
- Normal duoidal category (+)

*References*

- Commutativity (Garner, Lopez Franco, 2015)
- Duoidal Structures for Compositional Dependence (Shapiro, Spivak, 2022)

## # 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.

- A coherence theorem for duoidal categories can be found in the work of (Aguiar, Mahahan, 2009).

## # 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; (…)

## # 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.

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