symmetric monoidal category

Idea

• : a monoidal category whose tensor product is as commutative as possible
• There are different degrees to which higher categorical product may be commutative

Definition

• A symmetric monoidal category is a braided monoidal category for which the braiding
  • \(B_{x, y}: x \otimes y \to y \otimes x\)
  • satisfies the condition: \(B_{y, x} \circ B_{x, y} = 1_{x \otimes y}\) for all objects \(x, y\)
  • Intuitively, this means that switching things twice in the same direction has no effect.
• Expanding the definitions, we can also say a symmetric monoidal category is,
  • (Components) to begin with a category \(M\) equipped with:
    • tensor product: a functor \(\otimes: M \times M \to M\)
    • unit object: an object \(1 \in M\)
    • associator: natural isomorphism \(\alpha_{x, y, z}: (x \otimes y) \otimes z \to x \otimes (y \otimes z)\)
    • left unitor: a natural isomorphism \(\lambda_x: 1 \otimes x \to x\)
    • right unitor: a natural isomorphism \(\rho_x: x \otimes 1 \to x\)
    • braiding: a natural isomorphism \(B_{x, y}: x \otimes y \to y \otimes x\)
  • (Laws) We demand that
    • the associator obey the pentagon identity, which says this diagram commutes: $1
    • the associator and unitors obey the triangle identity, which says this diagram commutes: $2
    • the braiding and associator obey the first hexagon identity: $3
    • the braiding satisfies: \(B_{x, y} \circ B_{y, x} = 1_{x \otimes y}\) for all objects \(x, y\)

Properties

• ...