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Duality is a concept that pops up in different areas of mathematics as well as other science, but besides being a "woo isn't that nice?", is there anything more to duality (than loosely stated some type of isomorphic behaviour)?

Has anyone seen duality (trippliality?? or more) between more than two conceptual objects?

Edit : By "anything more" I mean a systematic study and categorisation of duality on its own, maybe a chapter in a book. For example the properties of holomorphic functions are studied separate of the specific holomorphic functions themselves, so one can look at holomorphic functions as a subject of study on it's own right. Looking for something similar regarding the concept of duality, not the specific objects having duality (or triality), can we have a definition of duality independent of specific examples? In other words can we say this is what is meant by a duality relationship (or triality) and show that specific examples actually satisfy the definition and hence they have duality relationship wrt each other?

Apologies for a very ill defined question, but I wouldn't even know what area does a question like this belongs to.

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    http://en.wikipedia.org/wiki/Triality2011-09-12
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    I don't understand what you mean by "anything more." What examples do you have in mind?2011-09-12
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    @Qiaochu : By anything more I mean a systematic study and catehorisation of duality on it's own, maybe a chapter in a book. For example the properties of holomorphic funstions are studied separate of the specific holomorphic funstions themselves, so one can look at holomorphic funstions as a subject of study itself. Looking for something similar regarding the concept of duality, but not the spefcific objects having duality (or triality), can we have a defintion of duality independent of specific examples?2011-09-12
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    @Arjang: I think "duality" has a sufficiently different meaning in enough different contexts that you're not going to be able to find an overarching theory of duality, although I would love to be proved wrong!2011-09-12
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    @Mike : If intuisticly we can recognise it (up to some isomorphisim) even a dumb robot should be able to do the same with some instructions, even some type of defintion for some speficfic dulaities would be better than nothing at all, don't need a universal definition but a local definition for some contexts would be nice to have.2011-09-12
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    For the modified question, perhaps http://mathoverflow.net/questions/73711/the-concept-of-duality will be useful.2011-09-12

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