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
for each triple of objects . Mac Lane's coherence theorem states that, provided the following diagram commutes for all quadruples of objects ,
any pair of morphisms from to constructed as compositions of various are equal.
- Mac Lane, Saunders (1971). "Categories for the working mathematician". Graduate texts in mathematics Springer-Verlag. Especially Chapter VII.