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dekeyser-kevin97 / Category Theory

May 17, 2025 16:25
category theory
compact closed categories
mathematics
monoidal categories
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Category Theory

A comprehensive cheat sheet covering monoidal category theory, its prerequisites, string diagrams, and compact closed categories. Useful for students and researchers in mathematics, physics, and computer science.

Prerequisites: Category Theory Basics

Categories

Definition: A category C consists of:

  • A class of objects, denoted ob(C).
  • For each pair of objects A, B, a set of morphisms (or arrows) Hom(A, B).
  • A composition law: for f: A → B and g: B → C, there exists a morphism g ∘ f: A → C.
  • For each object A, an identity morphism idA: A → A.

Axioms:

  • Associativity: h ∘ (g ∘ f) = (h ∘ g) ∘ f
  • Identity: f ∘ idA = f = idB ∘ f for f: A → B

Examples:

  • Set: Sets as objects, functions as morphisms.
  • Vect: Vector spaces as objects, linear transformations as morphisms.
  • Grp: Groups as objects, group homomorphisms as morphisms.

Functors: A functor F: C → D maps objects and morphisms from category C to category D preserving composition and identity.

Isomorphisms: A morphism f: A → B is an isomorphism if there exists a morphism g: B → A such that g ∘ f = idA and f ∘ g = idB.

Natural Transformations: A natural transformation α: F → G between functors F, G: C → D assigns to each object A in C a morphism αA: F(A) → G(A) in D such that for any morphism f: A → B in C, G(f) ∘ αA = αB ∘ F(f).

Products and Coproducts

Product: The product of objects A and B in a category C is an object A × B together with morphisms π₁: A × B → A and π₂: A × B → B such that for any object X and morphisms f: X → A and g: X → B, there exists a unique morphism h: X → A × B such that π₁ ∘ h = f and π₂ ∘ h = g.

Coproduct: The coproduct of objects A and B in a category C is an object A + B together with morphisms ι₁: A → A + B and ι₂: B → A + B such that for any object X and morphisms f: A → X and g: B → X, there exists a unique morphism h: A + B → X such that h ∘ ι₁ = f and h ∘ ι₂ = g.

Examples (Set):

  • Product: Cartesian product of sets.
  • Coproduct: Disjoint union of sets.

Examples (Vect):

  • Product: Direct product of vector spaces.
  • Coproduct: Direct sum of vector spaces.

Monoidal Categories

Monoidal Category Structure

Definition: A monoidal category (C, ⊗, I, α, λ, ρ) consists of:

  • A category C.
  • A bifunctor ⊗: C × C → C (tensor product).
  • An object I ∈ ob(C) (unit object).
  • A natural isomorphism α: (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C) (associator).
  • Natural isomorphisms λ: I ⊗ A → A and ρ: A ⊗ I → A (left and right unitors).

Axioms (Coherence):

  • Pentagon identity: A coherence condition for the associator.
  • Triangle identity: A coherence condition relating the associator, left unitor, and right unitor.

Examples of Monoidal Categories

  • (Set, ×, {∗}): Sets with Cartesian product and a singleton set as the unit.
  • (Vect, ⊗, K): Vector spaces over a field K with the tensor product and K itself as the unit.
  • (Cat, ×, 1): Categories with the Cartesian product and the terminal category as the unit.

Braided Monoidal Categories

Definition: A braided monoidal category is a monoidal category C with a natural isomorphism β: A ⊗ B → B ⊗ A (braiding) satisfying additional coherence axioms.

Examples:

  • Symmetric monoidal categories where β is an involution (β² = id).
  • The category of representations of a quantum group.

String Diagrams

Basic Conventions

  • Objects are represented by wires.
  • Morphisms are represented by boxes on the wires.
  • Composition is represented by connecting boxes vertically.
  • Tensor product is represented by placing wires side by side horizontally.

String Diagrams for Monoidal Categories

  • Associator α: (A ⊗ B) ⊗ C → A ⊗ (B ⊗ C) is a crossing.
  • Left unitor λ: I ⊗ A → A is a bending to the left.
  • Right unitor ρ: A ⊗ I → A is a bending to the right.
  • Identity id: A → A is a straight wire.

Examples

String diagrams provide a visual representation that makes complex compositions and tensor products easier to understand. For example, a diagram showing the composition of multiple morphisms in a monoidal category will clearly show the flow of data and how different parts of the diagram interact.

Compact Closed Categories

Definition and Properties

Definition: A compact closed category is a symmetric monoidal category where every object A has a dual object A** and morphisms ηA: I → A ⊗ A** (unit) and εA: A** ⊗ A → I (counit) satisfying the snake equations.

Snake Equations:

(εA ⊗ idA) ∘ (idA** ⊗ ηA) = idA**

(idA ⊗ εA**) ∘ (ηA ⊗ idA) = idA

Graphical Representation

  • The unit ηA is represented by a cup.
  • The counit εA is represented by a cap.
  • The snake equations are depicted as straightening a bent wire.

Examples

  • Hilb: Hilbert spaces with bounded linear operators.
  • Rel: Sets and relations.