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.
An illustrative example: a monoidal category
Part of the data of a monoidal category is a chosen morphism , called the associator:
for each triple of objects in the category. Using compositions of these , one can construct a morphism
Actually, there are many ways to construct a morphism from
as a composition of various . 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 know that the rest also hold. In the above example, one only needs to check that, for all quadruples of objects , the following diagram commutes
Two simple examples that illustrate the definition are as follows. Both are directly from the definition of a category.
Let f : A → B be a morphism of a category containing two objects A and B. Associated with these objects are the identity morphisms 1A : A → A and 1B : B → B. By composing these with f, we construct two morphisms:
- f o 1A : A → B, and
- 1B o f : A → B.
Both are morphisms between the same objects as f. We have, accordingly, the following coherence statement:
- f o 1A = f = 1B o f.
Associativity of composition
Let f : A → B, g : B → C and h : C → D 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:
- (h o g) o f : A → D, and
- h o (g o f) : A → D.
We have now the following coherence statement:
- (h o g) o f = h o (g o f).
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.
- Mac Lane, Saunders (1971). "Categories for the working mathematician". Graduate texts in mathematics Springer-Verlag. Especially Chapter VII Part 2.