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)
Jacob Lurie, Ultracategories, (pdf)
Francisco Marmolejo, Ultraproducts and continuous families of models, (thesis, poor quality photocopy).
Some more general variants are achieved in
Marek W. Zawadowski, Descent and duality, Annals of Pure and Applied Logic 71, n.2 (1995), 131–188
Henrik Forssell, First-order logical duality, Ph.D. thesis, Carnegie Mellon U. 2008, pdf