Coherence condition

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.

Part of the data of a monoidal category is a chosen morphism ${\displaystyle \alpha _{A,B,C}}$, called the associator:

${\displaystyle \alpha _{A,B,C}\colon (A\otimes B)\otimes C\rightarrow A\otimes (B\otimes C)}$

for each triple of objects ${\displaystyle A,B,C}$ in the category. Using compositions of these ${\displaystyle \alpha _{A,B,C}}$, one can construct a morphism

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