# Coherence theorem

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.