Coherence theorem

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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[edit]

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.png 

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[edit]

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