monoidal category

Last updated Dec 7, 2024

• Monoidal category
• Index

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

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
  • Cartesian monoidal category
    • uniform copy delete
    • cartesian categories are monoidal categories
    • cartesian categories and comonoids
• compact closed monoidal category
  • Duals and compact closed categories

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)