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
Riemann integration, Lebesgue integration
line integral/contour integration
integration of differential forms
integration over supermanifolds, Berezin integral, fermionic path integral
Kontsevich integral, Selberg integral, elliptic Selberg integral
integration in ordinary differential cohomology
integration in differential K-theory
A special kind of Poincaré duality space: a CW-complex which satisfies Poincaré duality and in this respect thus behaves like a closed manifold. Also called a homology manifold, therefore.
C. T. C. Wall, Poincaré complexes I, Annals of Mathematics, Second Series, Volume 86 (1967) pdf
Andrew Ranicki, 45 slides on chain duality, September 1996 (pdf)
With an eye towards generalization in spectral geometry (spectral triples):