Monoidal category

Last updated May 2, 2024

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

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 monoidal category

• cartesian categories are monoidal categories
• 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
• do-notation for symmetric monoidal categories
• distributive law
• String diagrams for monad and monad algebras
• Motivating bicategories

As theories of processes.

• process theory

Examples

• Monoidal category of dualities
• Monoidal category of streams

Literature.

• Categories for the Working Mathematician (MacLane)
• The Geometry of Tensor Calculus I (Joyal, Street)

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