Coherence condition

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In mathematics, a coherence condition or coherence theorem expresses the statement that two or more morphisms between two given objects, the existence of which is given or follows from general properties, are equal. Such situations are ubiquitous in mathematics. Coherence conditions and theorems are formulated and studied in the branch of mathematics known as category theory, at a level of abstraction beyond the everyday usage of most mathematicians.

1 Condition or theorem?
2 Examples

[edit] Condition or theorem?

The difference between the use of "condition" and "theorem" is as always in mathematics. If the statement in question is the conclusion of a proof, it is a theorem. If, on the other hand, the statement is one that is required to hold in order that certain conclusions may be drawn, it is a condition.

[edit] Examples

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

[edit] 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:

f o 1_A : A → B, and

1_B o f : A → B.

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

f o 1_A = f = 1_B o f.

[edit] 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:

(h o g) o f : A → D, and

h o (g o f) : A → D.

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.

It is, however, actually not common practice in category theory to refer to these particular elementary identities as being coherence statements, the term usually being reserved for less elementary situations, sometimes expressed in the form of a commuting diagram.