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| | 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. |
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| | ==Examples== |
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| | 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 |
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| | <math>\alpha_{A,B,C} \colon (A\otimes B)\otimes C \rightarrow A\otimes(B\otimes C)</math> |
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| | for each triple of objects <math>A,B,C</math>. Mac Lane's coherence theorem states that, provided the following diagram commutes for all quadruples of objects <math>A,B,C, D</math>, |
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| | [[Image:monoidal-category-pentagon.png]] |
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| | any pair of morphisms from <math> ( \cdots ( A_N \otimes A_{N-1} ) \otimes \cdots ) \otimes A_2 ) \otimes A_1) </math> to <math> ( A_N \otimes ( A_{N-1} \otimes \cdots \otimes ( A_2 \otimes A_1) \cdots ) </math> constructed as compositions of various <math>\alpha_{A,B,C}</math> are equal. |
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| | == References == |
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| | * [[Saunders Mac Lane|Mac Lane, Saunders]] (1971). "Categories for the working mathematician". ''Graduate texts in mathematics'' Springer-Verlag. Especially Chapter VII. |
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| | [[Category:Category theory]] |
Revision as of 06:12, 3 October 2012
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
for each triple of objects 
. Mac Lane's coherence theorem states that, provided the following diagram commutes for all quadruples of objects 
,
any pair of morphisms from 
to 
constructed as compositions of various 
are equal.
References
- Mac Lane, Saunders (1971). "Categories for the working mathematician". Graduate texts in mathematics Springer-Verlag. Especially Chapter VII.
