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 $$\alpha_{A,B,C}$$, called the associator:


 * $$\alpha_{A,B,C} \colon (A\otimes B)\otimes C \rightarrow A\otimes(B\otimes C)$$

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


 * $$( ( A_N \otimes A_{N-1} ) \otimes A_{N-2} ) \otimes \cdots \otimes A_1) \rightarrow ( A_N \otimes ( A_{N-1} \otimes \cdots \otimes ( A_2 \otimes A_1) ). $$

Actually, there are many ways to construct such a morphism as a composition of various $$\alpha_{A,B,C}$$. 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 $$A,B,C,D$$, the following diagram commutes.

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.

Further examples
Two simple examples that illustrate the definition are as follows. Both are directly from the definition of a category.

Identity
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.