11.4 Coherence theorems

Monoidal categories have special rules for combining objects. These rules, called coherence theorems, ensure that no matter how you group things, the result is the same. This makes working with monoidal categories much simpler.

Braided and symmetric monoidal categories add extra rules for swapping objects. Their coherence theorems make sure these swaps work consistently with the other rules. This allows for even more flexible ways of combining and rearranging objects.

Coherence Theorems for Monoidal Categories

Coherence theorem for monoidal categories

• States any well-formed diagram composed of associators and unitors commutes
  • Associators are isomorphisms $\alpha_{A,B,C}: (A \otimes B) \otimes C \to A \otimes (B \otimes C)$ that satisfy the pentagon identity
  • Unitors are isomorphisms $\lambda_A: I \otimes A \to A$ and $\rho_A: A \otimes I \to A$ that satisfy the triangle identity
• Implies any two parallel morphisms built from associators and unitors are equal
• Allows unambiguous interpretation of well-formed expressions involving the monoidal product without the need for parentheses (e.g., $(A \otimes B) \otimes C = A \otimes (B \otimes C)$)
• Simplifies manipulation of morphisms in monoidal categories by ensuring consistency (e.g., $(f \otimes g) \otimes h = f \otimes (g \otimes h)$ for morphisms $f, g, h$)

Proof of monoidal coherence theorem

• Relies on the following steps:
  1. Define a strict monoidal category $C_s$ with the same objects as the original monoidal category $C$
  2. Construct a monoidal functor $F: C \to C_s$ that is bijective on objects and full and faithful
  3. Show any diagram in $C$ commutes if and only if its image under $F$ commutes in $C_s$
  4. Prove any well-formed diagram in $C_s$ commutes using strict associativity and unit properties
• Functor $F$ is constructed by choosing a unique bracketing for each object and morphism in $C$ (e.g., $((A \otimes B) \otimes C) \otimes D$ for objects, $((f \otimes g) \otimes h) \otimes k$ for morphisms)
• Coherence theorem follows from the fact that any well-formed diagram in $C$ maps to a commutative diagram in $C_s$ under $F$

Coherence Theorems for Braided and Symmetric Monoidal Categories

Coherence theorem for braided monoidal categories

• States any well-formed diagram composed of associators, unitors, and braiding isomorphisms commutes
  • Braiding isomorphisms are natural isomorphisms $\gamma_{A,B}: A \otimes B \to B \otimes A$ that satisfy the hexagon identities
• Implies any two parallel morphisms built from associators, unitors, and braiding isomorphisms are equal
• Allows consistent manipulation of morphisms in braided monoidal categories, taking into account the braiding structure (e.g., $\gamma_{B,C} \circ (f \otimes g) = (g \otimes f) \circ \gamma_{A,B}$ for morphisms $f: A \to B$ and $g: C \to D$)

Comparison of coherence theorems

• Monoidal categories:
  • Coherence theorem involves associators and unitors
  • Ensures unambiguous interpretation of well-formed expressions involving the monoidal product (e.g., $(A \otimes B) \otimes C = A \otimes (B \otimes C)$)
• Braided monoidal categories:
  • Coherence theorem includes associators, unitors, and braiding isomorphisms
  • Ensures consistency in the presence of a braiding structure, which allows swapping of factors in the monoidal product (e.g., $A \otimes B \cong B \otimes A$ via $\gamma_{A,B}$)
• Symmetric monoidal categories:
  • A symmetric monoidal category is a braided monoidal category with an additional condition: $\gamma_{B,A} \circ \gamma_{A,B} = id_{A \otimes B}$
  • Coherence theorem is similar to that of braided monoidal categories but with the added condition on braiding isomorphisms
  • Braiding isomorphisms are involutive, meaning swapping factors twice returns to the original order (e.g., $\gamma_{B,A} \circ \gamma_{A,B} = id_{A \otimes B}$ implies $(B \otimes A) \otimes C \cong B \otimes (A \otimes C)$)