Coherence condition

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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[edit]

Part of the data of a monoidal category is a chosen morphism , called the associator:

for each triple of objects in the category. Using compositions of these , one can construct a morphism

Actually, there are many ways to construct such a morphism as a composition of various . 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 , the following diagram commutes.

Monoidal category pentagon.svg

Any pair of morphisms from to constructed as compositions of various are equal.

Further examples[edit]

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

Identity[edit]

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, and
1B 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[edit]

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, and
h 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[edit]

  • Mac Lane, Saunders (1971). Categories for the working mathematician. Graduate texts in mathematics Springer-Verlag. Especially Chapter VII Part 2.