Braided monoidal categories add a twist to regular monoidal categories. They introduce a braiding isomorphism that lets you swap objects in a tensor product, following specific rules to keep things consistent.

These categories pop up in quantum groups, braid theory, and anyons in physics. They're a step between regular monoidal categories and symmetric ones, offering more flexibility in how objects interact.

Braided Monoidal Categories

Definition of braided monoidal categories

• A braided monoidal category is a monoidal category $(\mathcal{C}, \otimes, I)$ equipped with a natural isomorphism called the braiding
  • The braiding is denoted as $\gamma_{A,B}: A \otimes B \to B \otimes A$ for all objects $A$ and $B$ in $\mathcal{C}$
  • The braiding isomorphism satisfies naturality and coherence conditions
• Naturality condition for the braiding: For any morphisms $f: A \to A'$ and $g: B \to B'$, the following diagram commutes:
  • $\gamma_{A',B'} \circ (f \otimes g) = (g \otimes f) \circ \gamma_{A,B}$
• The braiding isomorphism and its inverse $\gamma_{A,B}^{-1}: B \otimes A \to A \otimes B$ allow for swapping the order of objects in a tensor product

Coherence conditions for braiding isomorphisms

• The braiding isomorphism in a braided monoidal category must satisfy two coherence conditions known as the hexagon identities
  • Hexagon identity for $\gamma$: $(\mathrm{id}B \otimes \gamma{A,C}) \circ (\gamma_{A,B} \otimes \mathrm{id}C) = \gamma{A,B \otimes C}$
  • Hexagon identity for $\gamma^{-1}$: $(\gamma_{A,C} \otimes \mathrm{id}B) \circ (\mathrm{id}A \otimes \gamma{B,C}) = \gamma{A \otimes B,C}$
• These coherence conditions ensure consistency when multiple objects are involved in the tensor product
  • They guarantee that different ways of using the braiding to rearrange objects in a tensor product yield the same result
• Satisfying the hexagon identities is crucial for the well-definedness and consistency of the braided monoidal structure

Braided vs symmetric monoidal categories

• Braided and symmetric monoidal categories both have a braiding isomorphism $\gamma_{A,B}: A \otimes B \to B \otimes A$ that satisfies naturality and coherence conditions
• The key difference lies in an additional condition for symmetric monoidal categories:
  • In a symmetric monoidal category, the braiding isomorphism also satisfies $\gamma_{B,A} \circ \gamma_{A,B} = \mathrm{id}_{A \otimes B}$ for all objects $A$ and $B$
  • This condition implies that the braiding isomorphism is its own inverse, i.e., $\gamma_{A,B}^{-1} = \gamma_{B,A}$
• Every symmetric monoidal category is a braided monoidal category, but the converse is not true
  • There exist braided monoidal categories that are not symmetric, such as the category of representations of a non-cocommutative Hopf algebra

Examples of braided monoidal categories

• The category of representations of a quantum group $U_q(\mathfrak{g})$
  • Objects are representations of the quantum group, and morphisms are intertwiners between representations
  • The tensor product is given by the tensor product of representations
  • The braiding isomorphism arises from the quasi-triangular structure (R-matrix) of the quantum group
• The category of braids $\mathbf{Braid}$
  • Objects are natural numbers $n \in \mathbb{N}$ representing the number of strands
  • Morphisms are braids between $n$ and $m$ strands, with composition given by vertical stacking of braids
  • The tensor product is the disjoint union of braids, i.e., placing braids side by side
  • The braiding isomorphism is the crossing of two strands
• The category of $G$-graded vector spaces for a finite group $G$
  • Objects are vector spaces with a $G$-grading, and morphisms are grading-preserving linear maps
  • The tensor product is the tensor product of vector spaces with the induced $G$-grading
  • The braiding isomorphism is defined using the group multiplication in $G$

Applications of braiding isomorphisms

• Constructing new morphisms: Given morphisms $f: A \to A'$ and $g: B \to B'$ in a braided monoidal category, the braiding isomorphism allows creating morphisms like:
  • $\gamma_{A',B'} \circ (f \otimes g): A \otimes B \to B' \otimes A'$
  • $(g \otimes f) \circ \gamma_{A,B}: A \otimes B \to B' \otimes A'$
• Studying braid group representations and their connections to knot theory and low-dimensional topology
  • The category of braids $\mathbf{Braid}$ is a key example of a braided monoidal category
  • Braiding isomorphisms in $\mathbf{Braid}$ encode the essential structure of braids and their compositions
• Analyzing the properties of anyons in topological quantum field theories
  • Anyons are particles with exotic braiding statistics that can be described using braided monoidal categories
  • The braiding isomorphisms capture the non-trivial exchange behavior of anyons
• Investigating the structure of quantum groups and their representations
  • Braided monoidal categories provide a natural framework for studying quantum groups and their representation theory
  • The braiding isomorphisms in the category of representations of a quantum group encode important information about the quantum group itself