Coherence theorem

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In mathematics and particularly category theory, a coherence theorem is a tool for proving a coherence condition. Typically a coherence condition requires an infinite number of equalities among compositions of structure maps. A coherence theorem states that, in order to be assured that all these equalities hold, it suffices to check a small number of identities.


Consider the case of a monoidal category. Recall that part of the data of a monoidal category is an associator, which is a choice of morphism

\alpha_{A,B,C} \colon (A\otimes B)\otimes C \rightarrow A\otimes(B\otimes C)

for each triple of objects A,B,C. Mac Lane's coherence theorem states that, provided the following diagram commutes for all quadruples of objects A,B,C, D,

 Monoidal category pentagon.svg

any pair of morphisms from  ( ( \cdots ( A_N \otimes A_{N-1} ) \otimes \cdots ) \otimes A_2 ) \otimes A_1) to  ( A_N \otimes ( A_{N-1}  \otimes ( \cdots \otimes ( A_2 \otimes A_1) \cdots ) ) constructed as compositions of various \alpha_{A,B,C} are equal.