Coherence theorem - Alchetron, The Free Social Encyclopedia

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. A coherence theorem states that, in order to be assured that all these equalities hold, it suffices to check a small number of identities.

Examples

Consider the case of a monoidal category. Recall that part of the data of a monoidal category is an associator, which is a choice of morphism

α A , B , C : ( A ⊗ B ) ⊗ C → A ⊗ ( B ⊗ C )

for each triple of objects A , B , C . Mac Lane's coherence theorem states that, provided the following diagram commutes for all quadruples of objects A , B , C , D ,

any pair of morphisms from ( ( ⋯ ( A N ⊗ A N − 1 ) ⊗ ⋯ ) ⊗ A 2 ) ⊗ A 1 ) to ( A N ⊗ ( A N − 1 ⊗ ( ⋯ ⊗ ( A 2 ⊗ A 1 ) ⋯ ) ) constructed as compositions of various α A , B , C are equal.

References

Coherence theorem Wikipedia

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