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
See at Pontrjagin dual.
For infinity-groups:
Ulrich Bunke, Thomas Schick, Markus Spitzweck, Andreas Thom, Duality for topological abelian group stacks and T-duality, Proceedings of the ICM 2006 satellite conference, Valladolid, Spain, 2006. Zürich: EMS. Series of Congress Reports, 227-347 (2008) (arXiv:math/0701428)
Calder Daenzer, A groupoid approach to noncommutative T-duality, PhD, 2007 (pdf)