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 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, 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
Read more about this topic: Coherence Condition
Famous quotes containing the word category:
“Despair is typical of those who do not understand the causes of evil, see no way out, and are incapable of struggle. The modern industrial proletariat does not belong to the category of such classes.”
—Vladimir Ilyich Lenin (18701924)