🔢Category Theory Unit 11 – Monoidal Categories and Braiding

Key Concepts and Definitions

• Monoidal category consists of a category $C$, a bifunctor $\otimes: C \times C \to C$ called the tensor product, and an object $I$ called the unit object
• Associativity isomorphism $\alpha_{A,B,C}: (A \otimes B) \otimes C \to A \otimes (B \otimes C)$ satisfies the pentagon axiom
  • Pentagon axiom ensures consistency when reassociating objects in different ways
• Left and right unit isomorphisms $\lambda_A: I \otimes A \to A$ and $\rho_A: A \otimes I \to A$ satisfy the triangle axiom
  • Triangle axiom ensures compatibility between the unit object and the tensor product
• Braiding is a natural isomorphism $\gamma_{A,B}: A \otimes B \to B \otimes A$ satisfying the hexagon axioms
  • Hexagon axioms ensure consistency when braiding and associating objects
• Symmetric monoidal category has a braiding that satisfies $\gamma_{B,A} \circ \gamma_{A,B} = id_{A \otimes B}$
• Strict monoidal category has associativity and unit isomorphisms that are identities
• Monoidal functor is a functor between monoidal categories that preserves the monoidal structure

Monoidal Categories Explained

• Monoidal categories capture the notion of tensor products and parallel composition of morphisms
• Tensor product $\otimes$ allows combining objects and morphisms in a compatible way
  • $f \otimes g: A \otimes C \to B \otimes D$ for morphisms $f: A \to B$ and $g: C \to D$
• Unit object $I$ acts as an identity under the tensor product
  • $A \otimes I \cong A \cong I \otimes A$ for any object $A$
• Associativity and unit isomorphisms ensure coherence and consistency of the monoidal structure
• Monoidal categories generalize the notion of monoids from set theory to category theory
  • Monoids have an associative binary operation and an identity element
• Examples of monoidal categories include $(\mathbf{Set}, \times, 1)$, $(\mathbf{Vect}_k, \otimes, k)$, and $(\mathbf{Ab}, \otimes, \mathbb{Z})$
• Monoidal categories provide a framework for studying algebraic structures and quantum systems

Types of Monoidal Categories

• Strict monoidal categories have associativity and unit isomorphisms that are identities
  • $(A \otimes B) \otimes C = A \otimes (B \otimes C)$ and $I \otimes A = A = A \otimes I$
• Non-strict or weak monoidal categories have non-trivial associativity and unit isomorphisms
• Symmetric monoidal categories have a braiding satisfying $\gamma_{B,A} \circ \gamma_{A,B} = id_{A \otimes B}$
  • Braiding allows swapping the order of objects in the tensor product
• Braided monoidal categories have a braiding without the symmetry condition
• Cartesian monoidal categories have a tensor product given by the categorical product $\times$
  • Every object has a natural diagonal morphism $\Delta_A: A \to A \times A$
• Closed monoidal categories have an internal hom functor $[-,-]: C^{op} \times C \to C$
  • Internal hom satisfies an adjunction $C(A \otimes B, C) \cong C(A, [B, C])$
• *-autonomous categories are symmetric closed monoidal categories with a dualizing object

Braiding in Categories

• Braiding is a natural isomorphism $\gamma_{A,B}: A \otimes B \to B \otimes A$ satisfying the hexagon axioms
• Hexagon axioms ensure consistency between braiding and associativity
  • $\alpha_{B,A,C} \circ (\gamma_{A,B} \otimes id_C) \circ \alpha_{A,B,C} = (id_B \otimes \gamma_{A,C}) \circ \alpha_{B,C,A} \circ (\gamma_{A,B} \otimes id_C)$
  • $\alpha^{-1}_{A,B,C} \circ (id_A \otimes \gamma_{B,C}) \circ \alpha^{-1}_{A,C,B} = (\gamma_{A,C} \otimes id_B) \circ \alpha^{-1}_{C,A,B} \circ (id_C \otimes \gamma_{A,B})$
• Braiding allows swapping the order of objects in the tensor product
  • $\gamma_{A,B}: A \otimes B \to B \otimes A$ and $\gamma_{B,A}: B \otimes A \to A \otimes B$
• Symmetric braiding satisfies $\gamma_{B,A} \circ \gamma_{A,B} = id_{A \otimes B}$
• Braided monoidal categories model anyonic systems in quantum physics
  • Anyons are particles with exotic statistics beyond bosons and fermions
• Braiding is essential for studying knot invariants and braid groups in topology
• Braided tensor categories are used in the construction of quantum groups and Hopf algebras

Coherence Conditions and Diagrams

• Coherence conditions ensure consistency and compatibility of the monoidal structure
• Pentagon axiom for associativity: $\alpha_{A,B,C \otimes D} \circ \alpha_{A \otimes B, C, D} = (id_A \otimes \alpha_{B,C,D}) \circ \alpha_{A,B \otimes C, D} \circ (\alpha_{A,B,C} \otimes id_D)$
  • Ensures consistency when reassociating objects in different ways
• Triangle axiom for unit: $\rho_A \otimes id_B = (id_A \otimes \lambda_B) \circ \alpha_{A,I,B}$
  • Ensures compatibility between the unit object and the tensor product
• Hexagon axioms for braiding: $\alpha_{B,A,C} \circ (\gamma_{A,B} \otimes id_C) \circ \alpha_{A,B,C} = (id_B \otimes \gamma_{A,C}) \circ \alpha_{B,C,A} \circ (\gamma_{A,B} \otimes id_C)$ and $\alpha^{-1}_{A,B,C} \circ (id_A \otimes \gamma_{B,C}) \circ \alpha^{-1}_{A,C,B} = (\gamma_{A,C} \otimes id_B) \circ \alpha^{-1}_{C,A,B} \circ (id_C \otimes \gamma_{A,B})$
  • Ensure consistency between braiding and associativity
• Coherence theorems state that diagrams composed of associativity, unit, and braiding isomorphisms commute
  • Allows treating monoidal categories as if they were strict
• Graphical calculus represents morphisms as string diagrams
  • Composition is vertical stacking and tensor product is horizontal juxtaposition
• Coherence conditions have a geometric interpretation in terms of string diagrams

Applications and Examples

• Monoidal categories are used to model various algebraic and physical structures
• $(\mathbf{Set}, \times, 1)$ is a symmetric monoidal category with the Cartesian product as the tensor product
  • Models combining sets and functions in parallel
• $(\mathbf{Vect}_k, \otimes, k)$ is a symmetric monoidal category with the tensor product of vector spaces
  • Models combining linear maps and quantum systems
• $(\mathbf{Ab}, \otimes, \mathbb{Z})$ is a symmetric monoidal category with the tensor product of abelian groups
  • Models combining group homomorphisms and modules
• Braided monoidal categories model anyonic systems in quantum physics
  • Anyons are particles with exotic statistics beyond bosons and fermions
• Braided tensor categories are used in the construction of quantum groups and Hopf algebras
  • Quantum groups are deformations of classical Lie groups and Lie algebras
• Monoidal categories are used in the study of topological quantum field theories (TQFTs)
  • TQFTs assign algebraic data to manifolds and cobordisms
• Monoidal categories provide a framework for studying operad theory and higher category theory
  • Operads encode algebraic structures with multiple inputs and outputs

Related Structures and Generalizations

• Bicategories generalize monoidal categories by allowing morphisms between objects
  • Composition of 1-morphisms is associative up to 2-isomorphisms
• Tricategories further generalize bicategories by allowing morphisms between morphisms
  • Composition of 2-morphisms is associative up to 3-isomorphisms
• Higher categories continue this process, with $n$-morphisms and $(n+1)$-isomorphisms
• Monoidal $\infty$-categories are higher categorical analogues of monoidal categories
  • Defined using the language of quasi-categories or complete Segal spaces
• Braided monoidal $\infty$-categories incorporate braiding into the higher categorical setting
• Symmetric monoidal $\infty$-categories have a symmetric braiding in the higher categorical setting
• Enriched categories generalize monoidal categories by allowing hom-objects to live in another monoidal category
  • $\mathcal{V}$-enriched categories have hom-objects in a monoidal category $\mathcal{V}$
• Monoidal fibrations combine the notions of monoidal categories and fibrations
  • Fibrations are functors satisfying a lifting property for certain morphisms
• Hopf monoids are objects in a braided monoidal category with compatible multiplication and comultiplication
  • Generalize Hopf algebras to the categorical setting

Exercises and Problem-Solving Techniques

• Verify that a given category with a tensor product and unit object is a monoidal category
  • Check the associativity and unit axioms using commutative diagrams
• Prove that a given monoidal category is strict, braided, or symmetric
  • Show that the associativity and unit isomorphisms are identities or that the braiding satisfies the required conditions
• Construct examples of monoidal categories from familiar categories
  • Consider categories with products, coproducts, or tensor products as candidates for the monoidal structure
• Compute the braiding for specific objects in a braided monoidal category
  • Use the hexagon axioms and the properties of the tensor product and associativity
• Show that a given functor between monoidal categories is a monoidal functor
  • Verify that the functor preserves the tensor product and unit object up to natural isomorphisms
• Use graphical calculus to prove identities in a monoidal category
  • Represent morphisms as string diagrams and manipulate them using the axioms of the monoidal category
• Classify braidings in a given monoidal category
  • Determine if the braidings are symmetric or non-symmetric and if they are unique up to isomorphism
• Construct a braided monoidal category from a given strict monoidal category
  • Define the braiding using the tensor product and check the hexagon axioms
• Prove coherence theorems for monoidal categories using string diagrams or by induction on the structure of the diagrams
  • Show that any diagram composed of associativity, unit, and braiding isomorphisms commutes