Coherence Condition - An Illustrative Example: A Monoidal Category

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

( \cdots ( A_N \otimes A_{N-1} ) \otimes A_{N-2} ) \otimes \cdots \otimes A_1) \rightarrow ( A_N \otimes ( A_{N-1} \otimes \cdots \otimes ( A_2 \otimes A_1) \cdots ).

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


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