Mario Román


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Monoidal category

Last updated Jan 14, 2023

Monoidal categories are an algebraic structure for transformations that can be composed sequentially and in parallel.

# Definition

A monoidal category (MacLane), $(ℂ, ⊗, I, α, λ, ρ)$, is a category $ℂ$ equipped with a functor $(⊗)﹕ ℂ × ℂ → ℂ$, a unit $I \in ℂ$, and three natural isomorphisms: the associator $α_{X,Y,Z} ﹕ (X ⊗ Y) ⊗ Z ≅ X ⊗ (Y ⊗ Z)$, the left unitor $λ_{X} \colon I ⊗ X ≅ X$ and the right unitor $ρ_X﹕ X ⊗ I ≅ X$; such that the triangle and pentagon equations hold, $α_{X,I,Y} ⨾ (id_X ⊗ λ_{Y}) = ρ_{X} ⊗ id_Y$ and $(α_{X,Y,Z} ⊗ id) ⨾ α_{X,Y ⊗ Z,T} ⨾ (id_{X} ⊗ α_{Y,Z,T}) = α_{X ⊗ Y,Z,T} ⨾ α_{X,Y,Z ⊗ T}$.

A monoidal category is strict if $α$, $λ$ and $ρ$ are identities.

# See also



As theories of processes.