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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]]
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Latest revision as of 21:35, 25 November 2014
Hello from France. I'm glad to be here. My first name is Wayne.
I live in a town called Vertou in south France.
I was also born in Vertou 34 years ago. Married in June year 2007. I'm working at the college.
Here is my web blog ugg outlet