In category theory in mathematics, Mac Lane's coherence theorem, after Saunders Mac Lane, states that in any monoidal category, every well-formed diagram built from the associativity and unit isomorphisms commutes. The theorem can be stated as a strictification result, namely that every monoidal category is monoidally equivalent to a strict monoidal category.
Overview In a monoidal category, the tensor product is associative and unital only up to the natural isomorphisms given by the associator and the left and right unitors. Because these isomorphisms can be composed in many different ways, there are in principle many different arrows between the same tensor expressions. The central question is whether all these “canonical” arrows agree with one another. Mac Lane's coherence theorem gives a positive answer. It states that every diagram that can be built from the associativity and unit constraints necessarily commutes. This means that any two canonical morphisms obtained by rebracketing tensor products (using the associator and its inverse) or by inserting and removing the unit object (using the unitors and their inverses) are equal whenever they have the same source and target. As a result, the associator and unitors behave coherently, and no further identities beyond the pentagon and triangle need to be imposed. One important consequence is that every monoidal category is equivalent, in a sense appropriate to monoidal structure, to a strict monoidal category in which the associativity and unital laws hold on the nose. This “strictification” property explains why monoidal categories can often be treated as though they were strictly associative without loss of generality.
Background A monoidal category ( C , ⊗ , I ) {\displaystyle ({\mathcal {C}},\otimes ,I)} is a category equipped with a tensor product functor ⊗ : C × C → C {\displaystyle \otimes :{\mathcal {C}}\times {\mathcal {C}}\to {\mathcal {C}}} and a unit object I {\displaystyle I} , together with natural isomorphisms that express how the tensor product behaves. The tensor product is not strictly associative or strictly unital; instead, the structure includes three canonical natural isomorphisms:
the associator α A , B , C : ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C ) {\displaystyle \alpha _{A,B,C}:(A\otimes B)\otimes C\to A\otimes (B\otimes C)} ,
the left unitor λ A : I ⊗ A → A {\displaystyle \lambda _{A}:I\otimes A\to A} , and
the right unitor ρ A : A ⊗ I → A {\displaystyle \rho _{A}:A\otimes I\to A} .
These isomorphisms witness associativity and unitality “up to isomorphism”. Because different sequences of rebracketing and unit insertions can connect the same tensor expressions, there may be many canonical morphisms between the same source and target object. A central question is whether all such canonical morphisms agree with one another. To control this ambiguity, monoidal categories are required to satisfy two compatibility conditions, the pentagon and triangle identities, which relate the associator and unitors. These conditions are the starting point for Mac Lane’s coherence theorem.
Statement Let ( C , ⊗ , I , α , λ , ρ ) {\displaystyle ({\mathcal {C}},\otimes ,I,\alpha ,\lambda ,\rho )} be a monoidal category whose structural natural isomorphisms satisfy the pentagon and triangle identities. A formal tensor expression is any expression obtained from objects of
C {\displaystyle {\mathcal {C}}} using ⊗ {\displaystyle \otimes } and I {\displaystyle I} . A formal composite is any morphism obtained by finite compositions and tensor products of the morphisms
α A , B , C , α A , B , C − 1 , λ A , λ A − 1 , ρ A , ρ A − 1 , {\displaystyle \alpha _{A,B,C},\qquad \alpha _{A,B,C}^{-1},\qquad \lambda _{A},\qquad \lambda _{A}^{-1},\qquad \rho _{A},\qquad \rho _{A}^{-1},}
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