11.1 Definition and examples of monoidal categories

Monoidal categories extend regular categories by adding a tensor product and unit object. These new elements allow us to combine objects and morphisms in interesting ways, creating richer structures.

The tensor product lets us multiply objects, while the unit object acts like a neutral element. This setup mirrors familiar mathematical structures, making monoidal categories a powerful tool for understanding various algebraic systems.

Monoidal Categories

Components of monoidal categories

• Objects the objects of the underlying category $\mathcal{C}$
• Morphisms the morphisms of the underlying category $\mathcal{C}$
• Tensor product a bifunctor $\otimes: \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ assigns to each pair of objects $(A, B)$ an object $A \otimes B$
  • Also defined on morphisms for $f: A \to A'$ and $g: B \to B'$, there is a morphism $f \otimes g: A \otimes B \to A' \otimes B'$
• Unit object an object $I \in \mathcal{C}$ serves as an identity for the tensor product
• Coherence conditions the tensor product and unit object satisfy
  • Associativity a natural isomorphism $\alpha_{A,B,C}: (A \otimes B) \otimes C \to A \otimes (B \otimes C)$ for all objects $A, B, C$
  • Left unit a natural isomorphism $\lambda_A: I \otimes A \to A$ for all objects $A$
  • Right unit a natural isomorphism $\rho_A: A \otimes I \to A$ for all objects $A$

Examples of monoidal categories

• Category of sets $\mathbf{Set}$ with the Cartesian product $\times$ as the tensor product and the singleton set $\{*\}$ as the unit object
  • For sets $A$ and $B$, the Cartesian product $A \times B$ is the set of ordered pairs $(a, b)$ with $a \in A$ and $b \in B$
  • The singleton set $\{*\}$ satisfies $A \times \{*\} \cong A \cong \{*\} \times A$ for all sets $A$
• Category of vector spaces $\mathbf{Vect}_K$ over a field $K$ with the tensor product $\otimes_K$ and the field $K$ as the unit object
  • For vector spaces $V$ and $W$, the tensor product $V \otimes_K W$ is the vector space generated by elements of the form $v \otimes w$ with $v \in V$ and $w \in W$, subject to certain relations
  • The field $K$ satisfies $V \otimes_K K \cong V \cong K \otimes_K V$ for all vector spaces $V$

Constraints in monoidal categories

• Associativity constraint the natural isomorphism $\alpha_{A,B,C}: (A \otimes B) \otimes C \to A \otimes (B \otimes C)$ ensures the tensor product is associative up to isomorphism
  • Allows for unambiguous expressions like $A \otimes B \otimes C$ without specifying parentheses
  • Satisfies the pentagon identity, which ensures coherence when reassociating tensor products of four objects
• Unit constraints the natural isomorphisms $\lambda_A: I \otimes A \to A$ and $\rho_A: A \otimes I \to A$ ensure the unit object $I$ behaves as an identity for the tensor product
  • Allows for expressions like $I \otimes A$ and $A \otimes I$ to be identified with $A$
  • Satisfy the triangle identity, which ensures coherence when combining the associativity and unit constraints

Interactions in monoidal categories

• Tensor product is a bifunctor compatible with composition of morphisms
  • For morphisms $f: A \to A'$, $g: B \to B'$, $h: A' \to A''$, and $k: B' \to B''$, we have $(h \otimes k) \circ (f \otimes g) = (h \circ f) \otimes (k \circ g)$
• Unit object $I$ interacts with morphisms
  • For any morphism $f: A \to B$, there are morphisms $f \otimes \mathrm{id}_I: A \otimes I \to B \otimes I$ and $\mathrm{id}_I \otimes f: I \otimes A \to I \otimes B$
  • These morphisms are compatible with the unit constraints $\rho_B \circ (f \otimes \mathrm{id}_I) = f \circ \rho_A$ and $\lambda_B \circ (\mathrm{id}_I \otimes f) = f \circ \lambda_A$