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I am currently learning both about operator algebras and algebraic geometry. It seems as though in these 2 subjects (and many more I've come across from online reading), there is a strong notion of equivalence between geometric spaces and algebraic things (typically rings/modules over certain types of functions) (e.g. Gelfand Theorem, Affine Spaces with their suitable ring of normal functions, Serre-Swan etc). My question is if there is any more general theory in which these types of dualities between geometric spaces (or objects: vector bundles ~ modules) and algebraic structures coming from things on those spaces, are special cases?

p.s. I don't know the relevant tags for this question, so feel free to edit

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    There's an extremely general context for this called Isbell duality. You might enjoy Lawvere's thoughts on what he views as an attempt by Isbell to guarantee the category of sets allows as much such duality as possible, here: http://at.yorku.ca/t/o/p/d/65.htm – 2017-01-20

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