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# produoidal category

Last updated Jul 23, 2024

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## # Intuition

Produoidal categories are the profunctorial counterpart of duoidal categories. Every produoidal category can be normalized to a normal produoidal. The cofree produoidal category over a category is the monoidal spliced arrow category, and its normalization is the category of monoidal contexts. The notion of representable homomorphism between produoidal categories is that of produoidal functor.

### # Definition

Definition. A produoidal category is a category $ℂ$ endowed with two promonoidal structures, $$ℂ_⊗(X,Y;Z), ℂ_⊗(X), \mbox{ and } ℂ_{◁}(X,Y;Z), ℂ_◁(X),$$ such that one laxly distributes over the other. This is to say that it is endowed with the following laxators $$\left(∫^{Z,Z’} ℂ_◁(X,Y;Z) × ℂ_◁(X’,Y’;Z’) × ℂ_⊗(Z,Z’;W) \right) → \left( ∫^{U,U’} ℂ_⊗(X,X’;U) × ℂ_⊗(Y,Y’;U’) × ℂ_◁(U,U’;W) \right),$$ $$ℂ_{⊗}(X) → \left(∫^{Y_1,Y_2} ℂ_{⊗}(Y_1) × ℂ_{⊗}(Y_2) × ℂ_◁(Y_1,Y_2;X)\right),$$ $$\left(∫^{X_1,X_2} ℂ_{◁}(X_1) × ℂ_{◁}(X_2) × ℂ_⊗(X_1,X_2;Y)\right) → ℂ_{◁}(Y),$$ $$ℂ_{⊗}(X) → ℂ_{◁}(X).$$ Moreover, the coherence transformations of the second promonoidal structure must preserve this laxity.

### # Produoidal categories as virtual duoidal categories

Produoidal categories can be seen as virtual duoidal categories where morphisms factor. In this case, the two different directions of laxity conflict, and we may want to consider them as opvirtual duoidal categories.

### # Normal produoidal categories

A normal produoidal category is a produoidal category where $𝕍(•;N)≅𝕍(•;I)$.

### # Laxators in different notations

$$ψ_2 ﹕ 𝕍(•;(•_1⊗•_2)◁(•_3⊗•_4)) → 𝕍(•;(•_1◁•_3)⊗(•_2◁•_4))$$$$ψ_2 \colon 𝕍(•;•_1⊗•_2) × 𝕍(•_1;•◁•) × 𝕍(•_2;•◁•) → 𝕍(•;•_1◁•_2) × 𝕍(•_1;•⊗•) × 𝕍(•_2;•⊗•).$$ $$ψ_0﹕ 𝕍(•;I) → 𝕍(•;I◁I),$$ $$φ_0﹕ 𝕍(•;N⊗N) → 𝕍(•;N),$$