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.