Coherence condition

Last updated January 02, 2024

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 1_A : A → A and 1_B : B → B. By composing these with f, we construct two morphisms:

fo 1_A : A → B, and

1_Bof : A → B.

Both are morphisms between the same objects as f. We have, accordingly, the following coherence statement:

fo 1_A = f = 1_Bof.

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:

(hog) of : A → D, and

ho (gof) : A → D.

We have now the following coherence statement:

(hog) of = ho (gof).

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.

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References

Mac Lane, Saunders (1971). "7. Monoids §2 Coherence". Categories for the working mathematician. Graduate texts in mathematics. Vol. 4. Springer. pp. 161–165. doi:10.1007/978-1-4612-9839-7_8. ISBN 9781461298397.

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