Monoidal category

Last updated Jan 14, 2023

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

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.

Motivation.

• Motivating monoidal categories
• 1-Dimensional calculus
• Interchange law
• Coherence for monoidal categories
• Sets is a monoidal category
• Strict monoidal categories and coherence

Constructions

• Cartesian Categories are monoidal
• Cartesian Categories and comonoids
• Braided monoidal category
• Duals and compact closed categories
• Uniform copy delete
• Examples of monoidal category
• String diagrams for category theory
• Dualities
• Three notations for symmetric monoidal categories
• Distributive law
• String diagrams for monad and monad algebras
• Motivating bicategories

As theories of processes.

• Process theory

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