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.

Examples

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 $A,B,C,D$ ,

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.

References

Mac Lane, Saunders (1971). "Categories for the working mathematician". Graduate texts in mathematics Springer-Verlag. Especially Chapter VII.

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