Trisha Shetty (Editor)

Coherence condition

Updated on
Edit
Like
Comment
Share on FacebookTweet on TwitterShare on LinkedInShare on Reddit

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.

Contents

An illustrative example: a monoidal category

Part of the data of a monoidal category is a chosen morphism α A , B , C , called the associator:

α A , B , C : ( A B ) C A ( B C )

for each triple of objects A , B , C in the category. Using compositions of these α 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 α 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

Further examples

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

Identity

Let f : AB be a morphism of a category containing two objects A and B. Associated with these objects are the identity morphisms 1A : AA and 1B : BB. By composing these with f, we construct two morphisms:

f o 1A : AB, and1B o f : AB.

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 : AB, g : BC and h : CD 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 : AD, andh o (g o f) : AD.

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.

References

Coherence condition Wikipedia


Similar Topics