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In mathematics and physics, there are many notions of duality that aren't always similar. For example, the notion of the dual of a vector space seems wildly different from the Hodge dual of linear forms, which is vastly different from the duality between electric and magnetic fields in electrodynamics, which is different from the dualities in AdS/CFT in physics.

I was talking to a friend and they mentioned that perhaps there was a rigorous notion of duality within more abstract branches of mathematics that could encompass all of these notions at once, perhaps some construction within Category Theory. Is there anything of the sort?

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  • $\begingroup$ It's worth noting that Hodge duality is just a special case (for inner product spaces) of Poincaré duality, which operates between dual exterior powers/algebras of dual vector spaces. So those two examples are actually closely related. $\endgroup$
    – blargoner
    Commented May 5 at 4:02

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Regarding the purely mathematical point of view I think that a good starting point may be:

A duality of a category $\mathscr{C}$ is an antiequivalence $\ast:\mathscr{C}\to \mathscr{C}$ such that $\ast\ast\cong \text{id}_{\mathscr{C}}$.

But I'm by no means the most qualified person to answer your question, so take this as a purely personal opinion.

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