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.

[edit] 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

$\alpha_{A,B,C}$

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 $($ to $($ constructed a compositions of various $α A, B, C$ are equal.

[edit] References

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