- Duoid in a duoidal
- Bimonoid in a duoidal
- Strings for duoidal categories
- Compositional dependencies with duoidals
- Produoidal category (+)
- Normal duoidal category (+)
- Commutativity (Garner, Lopez Franco, 2015)
- Duoidal Structures for Compositional Dependence (Shapiro, Spivak, 2022)
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).
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.
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.