Coherence theorem

In mathematics, and particularly category theory a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category. A coherence theorem states that, in order to be assured that all these equalities hold, it suffices to check a small number of identities. In mathematics, and particularly category theory a coherence condition is a collection of conditions requiring that various compositions of elementary morphisms are equal. Typically the elementary morphisms are part of the data of the category. A coherence theorem states that, in order to be assured that all these equalities hold, it suffices to check a small number of identities. Part of the data of a monoidal category is a chosen morphism α A , B , C {displaystyle alpha _{A,B,C}} , called the associator: for each triple of objects A , B , C {displaystyle A,B,C} in the category. Using compositions of these α A , B , C {displaystyle alpha _{A,B,C}} , one can construct a morphism Actually, there are many ways to construct such a morphism as a composition of various α A , B , C {displaystyle alpha _{A,B,C}} . One coherence condition that is typically imposed is that these compositions are all equal. Typically one proves a coherence condition using a coherence theorem, which states that one only needs to check a few equalities of compositions in order to show that the rest also hold. In the above example, one only needs to check that, for all quadruples of objects A , B , C , D {displaystyle A,B,C,D} , the following diagram commutes. Any pair of morphisms from ( ( ⋯ ( A N ⊗ A N − 1 ) ⊗ ⋯ ) ⊗ A 2 ) ⊗ A 1 ) {displaystyle ((cdots (A_{N}otimes A_{N-1})otimes cdots )otimes A_{2})otimes A_{1})} to ( A N ⊗ ( A N − 1 ⊗ ( ⋯ ⊗ ( A 2 ⊗ A 1 ) ⋯ ) ) {displaystyle (A_{N}otimes (A_{N-1}otimes (cdots otimes (A_{2}otimes A_{1})cdots ))} constructed as compositions of various α A , B , C {displaystyle alpha _{A,B,C}} are equal.