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
(also nonabelian homological algebra)
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
What is called Verdier duality is the refinement of Poincaré duality from ordinary cohomology to abelian sheaf cohomology. The Grothendieck six operations formalism is a formalization of aspects of Verdier duality.
In general abstract formulation one says that a locally compact site $X$ (e.g. a locally compact topological space) satisfies Verdier duality if there exists a derived right adjoint $\mathbb{L}p^!$ (exceptional inverse image) to the operation $\mathbb{R}p_!$ of direct image with compact support for the terminal map $X \to \ast$. More generally, in the relative situation, for $f \colon X \to Y$ a map, Verdier duality of $f$ means that $\mathbb{R}f_!$ has a derived right adjoint. If here $X$ is a scheme and $f_! \simeq f_\ast$ coincides with the direct image (“Grothendieck context”) then this is called Grothendieck duality. Therefore one often speaks of Verdier-Grothendieck duality.
If these adjoints exist, then one has the dualizing object in a closed category (dualizing module) $\omega \coloneqq \mathbb{L}p^! k$ (the image under the extra right adjoint of the unit object) and for every abelian sheaf $V$ on $X$ can define the dual object in a closed category
The analog of Poincaré duality is then the statement that the abelian sheaf cohomology with coefficients in $V$ is dual to the abelian sheaf cohomology with compact support with coefficients in $\mathbb{D}V$.
A fairly general class of implementations of Verdier-Grothendieck duality is discussed in (Joshua, theorem 1.3).
With assumption (Joshua, 4.3) this derives that dualization intertwines the two pushforwards as $\mathbb{D} \circ f_\ast \simeq f_! \circ \mathbb{D}$ (Joshua, corollary 5.4).
The duality is named after Jean-Louis Verdier.
Accounts of the traditional notion include
Wikipedia, Verdier duality
Liviu Niculaescu, The derived category of sheaves and the Poincaré-Verdier duality (pdf)
Jacob Lurie, lecture notes on Verdier duality (2011 pdf, 2013 pdf, 2014 pdf)
A general abstract formalization is in
Discussion in the context of higher algebra/higher geometry is in
Roy Joshua, Generalised Verdier duality for presheaves of spectra—I, Journal of Pure and Applied Algebra Volume 70, Issue 3, 29 March 1991, Pages 273–289
Jonathan Block and Andrey Lazarev, Homotopy Theory and Generalized Duality for Spectral Sheaves, IMRN International Mathematics Research Notices
1996, No. 20 (pdf)
Vladimir Drinfeld, Dennis Gaitsgory, section 5.3 On some finiteness questions for algebraic stacks (arXiv:1108.5351)
Jacob Lurie, section 4.2 of Representability theorems
Formulation in terms of an equivalence between sheaves and cosheaves is in
Peter Schneider, Verdier duality on the building, Journal reine angew. Math. 494, 1998
Justin Curry, Sheaves, Cosheaves and Applications (arXiv:1303.3255)
Jacob Lurie, section 5.5.5 of Higher Algebra
Discussion in the context of prestacks and diagrams of schemes
Dennis Gaitsgory, The Atiyah-Bott formula for the cohomology of the moduli space of bundles on a curve (arXiv:1505.02331)
Alexey Kalugin, On categorical approach to Verdier Duality (arXiv:1505.06922)