# 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
, 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.

Any pair of morphisms from to constructed as compositions of various 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:

*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*.

### 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.

## References

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