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
$\alpha_$, called the ''associator'':

: $\alpha_ \colon (A\otimes B)\otimes C \rightarrow A\otimes(B\otimes C)$

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

: $( ( A_N \otimes A_ ) \otimes A_ ) \otimes \cdots \otimes A_1) \rightarrow ( A_N \otimes ( A_ \otimes \cdots \otimes ( A_2 \otimes A_1) ).$

Actually, there are many ways to construct such a morphism as a composition of various $\alpha_$. 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 $( ( \cdots ( A_N \otimes A_ ) \otimes \cdots ) \otimes A_2 ) \otimes A_1)$ to $( A_N \otimes ( A_ \otimes ( \cdots \otimes ( A_2 \otimes A_1) \cdots ) )$ constructed as compositions of various $\alpha_$ 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

* {{cite book , author-link=Saunders Mac Lane , last=Mac Lane , first=Saunders , date=1971 , title=Categories for the working mathematician , series=Graduate texts in mathematics , volume=4 , publisher=Springer , chapter=7. Monoids §2 Coherence , pages=161–165 , title-link=Categories for the Working Mathematician , doi=10.1007/978-1-4612-9839-7_8 , chapter-url=https://link.springer.com/chapter/10.1007/978-1-4612-9839-7_8

Category theory