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. A coherence theorem states that, in order to be assured that all these equalities hold, it suffices to check a small number of identities.
An illustrative example: a monoidal category.
Part of the data of a monoidal category is a chosen morphism
formula_1, called the "associator":
for each triple of objects formula_3 in the category. Using compositions of these formula_1, one can construct a morphism
Actually, there are many ways to construct such a morphism as a composition of various formula_1. 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 formula_7, the following diagram commutes.
Any pair of morphisms from formula_8 to formula_9 constructed as compositions of various formula_1 are equal.
Further examples.
Two simple examples that illustrate the definition are as follows. Both are directly from the definition of a category.
Identity.
Let be a morphism of a category containing two objects "A" and "B". Associated with these objects are the identity morphisms and . By composing these with "f", we construct two morphisms:
Both are morphisms between the same objects as "f". We have, accordingly, the following coherence statement:
Associativity of composition.
Let , and be morphisms of a category containing objects "A", "B", "C" and "D". By repeated composition, we can construct a morphism from "A" to "D" in two ways:
We have now the following coherence statement:
In these two particular examples, the coherence statements are "theorems" for the case of an abstract category, since they follow directly from the axioms; in fact, they "are" axioms. For the case of a concrete mathematical structure, they can be viewed as conditions, namely as requirements for the mathematical structure under consideration to be a concrete category, requirements that such a structure may meet or fail to meet.