abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
Makkai duality is a kind of syntax-semantics duality, due to Michael Makkai, relating pretoposes (categories having to do with the syntax of first-order logic) and ultracategories (which are a way of capturing the semantics of first-order logic). This leads to a proof of conceptual completeness for first-order logic.
Michael Makkai, Stone duality for first order logic, Proceedings of the Herbrand symposium (Marseilles, 1981), 217–232, Stud. Logic Found. Math. 107, North-Holland, Amsterdam, 1982, doi;
Michael Makkai, Stone duality for first order logic, Adv. Math. 65 (1987) no. 2, 97–170, doi, MR89h:03067;
Michael Makkai, Duality and definability in first order logic, Mem. Amer. Math. Soc. 105 (1993), no. 503
Jacob Lurie, Lecture 29X-Makkai Duality, (lecture notes)
Some more general variants are achieved in
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