Monoidal category

Last updated Feb 15, 2025

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

• Categories for the Working Mathematician (MacLane)

References.

• Categories for the Working Mathematician (MacLane)
• Extracting a proof of coherence for monoidal categories from a formal proof of normalization for monoids (Beylin, Dybjer)

• cartesian categories are monoidal categories
• Fox’s theorem and cocommutativity
• Refining Fox’s Theorem
• uniform copy delete
• Cartesian by split
• Cartesian, partial and stochastic

Motivating monoidal categories

• 1-Dimensional calculus
• interchange law
• Coherence for monoidal categories
• Sets is a monoidal category
• Strict monoidal categories and coherence
• Examples of monoidal category
• String diagrams for category theory
• Three notations for symmetric monoidal categories
• Lax monoidal functor

Types.

• braided monoidal category
• symmetric monoidal category
  • copy-discard category
    • copy-discard-compare category
  • Markov category
    • partial Markov category
    • discrete partial Markov category
    • cartesian restriction category
• compact closed monoidal category
  • Duals and compact closed categories

• dialogue category

string diagrams

• String diagrams for monad and monad algebras
• string diagrams for bicategories
• string diagrams are two

Constructions

• Dualities
• do-notation - theory of 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)