Coherence condition explained

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 ⊗ B) ⊗ C → A ⊗ (B ⊗ C) A, B, C in the category. Using compositions of these \alpha _{A,B,C} , one can construct a morphism( ( A _{N} ⊗ A _{N-1} ) ⊗ A _{N-2} ) ⊗ … ⊗ A _{1)} → ( A _{N} ⊗ ( A _{N-1} ⊗ … ⊗ ( A _{2} ⊗ 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

( ( … ( A _{N} ⊗ A _{N-1} ) ⊗ … ) ⊗ A _{2} ) ⊗ A _{1)} to ( A _{N} ⊗ ( A _{N-1} ⊗ ( … ⊗ ( A _{2} ⊗ A _{1)} … ) ) 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 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:

, and

.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:

, and

.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.

References

• Book: Mac Lane, Saunders . Saunders Mac Lane . 1971 . Categories for the working mathematician . Graduate texts in mathematics . 4 . Springer . 7. Monoids §2 Coherence . 161–165 . Categories for the Working Mathematician . 10.1007/978-1-4612-9839-7_8 . 9781461298397 . https://link.springer.com/chapter/10.1007/978-1-4612-9839-7_8.