In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, and particularly category theory

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

 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.

An illustrative example: a monoidal category

Monoidal category

In mathematics, a monoidal category is a category C equipped with a bifunctorwhich is associative, up to a natural isomorphism, and an object I which is both a left and right identity for ⊗, again up to a natural isomorphism...

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 a morphism from



to



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

Coherence theorem

In mathematics and particularly category theory, a coherence theorem is a tool for proving a coherence condition. Typically a coherence condition requires an infinite number of equalities among compositions of structure maps...

, which states that one only needs to check a few equalities of compositions in order to know that the rest also hold. In the above example, one only needs to check that, for all quadruples of objects , 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 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.