Symmetric monoidal categories expand on monoidal categories by adding a symmetry isomorphism. This allows objects in tensor products to swap order, maintaining consistency with associativity and unit isomorphisms. It's like having a commutative tensor product, but with more flexibility.

Examples include sets with Cartesian products, vector spaces with tensor products, and abelian groups. These categories find applications in rearranging tensor products, defining commutativity, and constructing braided monoidal categories, which are useful in knot theory and quantum field theory.

Symmetric Monoidal Categories

Definition of symmetric monoidal categories

• Extends the concept of a monoidal category $(C, \otimes, I)$ by adding a natural isomorphism called the symmetry isomorphism
  • For any two objects $A$ and $B$ in $C$, there exists an isomorphism $\sigma_{A,B}: A \otimes B \to B \otimes A$ that swaps the order of the objects in the tensor product
  • The symmetry isomorphism must satisfy certain coherence conditions to ensure consistency with the associativity and unit isomorphisms of the monoidal category
• Captures the idea of a tensor product that is commutative up to isomorphism (e.g., the tensor product of vector spaces or the Cartesian product of sets)

Coherence conditions for symmetry isomorphisms

• Compatibility with the associativity isomorphism: $\sigma_{A,B \otimes C} = (id_B \otimes \sigma_{A,C}) \circ (\sigma_{A,B} \otimes id_C)$ and $\sigma_{A \otimes B,C} = (\sigma_{A,C} \otimes id_B) \circ (id_A \otimes \sigma_{B,C})$
  • Ensures that the symmetry isomorphism behaves consistently when applied to nested tensor products
• Involutive property: $\sigma_{B,A} \circ \sigma_{A,B} = id_{A \otimes B}$
  • Applying the symmetry isomorphism twice in succession yields the identity morphism, effectively canceling out the swapping of objects

Examples of symmetric monoidal categories

• The category of sets $(Set, \times, {})$ with the Cartesian product as the tensor product and a singleton set as the unit object
  • The symmetry isomorphism is given by the function $\sigma_{A,B}(a,b) = (b,a)$ for sets $A$ and $B$, which swaps the order of elements in the Cartesian product
• The category of vector spaces $(Vect_k, \otimes, k)$ over a field $k$ with the tensor product and the base field as the unit object
  • The symmetry isomorphism is given by $\sigma_{V,W}(v \otimes w) = w \otimes v$ for vectors $v \in V$ and $w \in W$, which swaps the order of vectors in the tensor product
• The category of abelian groups $(Ab, \otimes, \mathbb{Z})$ with the tensor product and the integers as the unit object
  • The symmetry isomorphism is given by $\sigma_{A,B}(a \otimes b) = b \otimes a$ for abelian groups $A$ and $B$, which swaps the order of elements in the tensor product

Applications of symmetry isomorphisms

• Rearranging the order of objects in a tensor product
  • Given morphisms $f: A \to C$ and $g: B \to D$, we can define a morphism $g \otimes f: B \otimes A \to D \otimes C$ as $(g \otimes f) = (\sigma_{C,D} \circ (f \otimes g)) \circ \sigma_{A,B}$
  • This allows for the composition of morphisms in a different order, which can simplify calculations or proofs
• Defining the notion of commutativity for objects in a symmetric monoidal category
  • An object $A$ is commutative if $\sigma_{A,A} = id_{A \otimes A}$, meaning that the symmetry isomorphism applied to $A \otimes A$ is the identity morphism
  • Commutative objects have the property that the order of elements in the tensor product does not matter (e.g., the tensor product of two commutative rings is commutative)
• Constructing braided monoidal categories, which generalize symmetric monoidal categories by relaxing the involutive property of the symmetry isomorphism
  • Braided monoidal categories have applications in knot theory and topological quantum field theory