abstract duality: opposite category,
concrete duality: dual object, dualizable object, fully dualizable object, dualizing object
Examples
between higher geometry/higher algebra
Langlands duality, geometric Langlands duality, quantum geometric Langlands duality
In QFT and String theory
The stable (infinity,1)-category of spectra has a dualizing object (dualizing module) on a suitable subcategory of finite spectra. It is called the Anderson spectrum $I_{\mathbb{Z}}$ (Lurie, Example 4.3.9). The duality that this induces is called Anderson duality.
The Anderson dual of the sphere spectrum is discussed in
(Hopkins-Singer 05, appendix B) in the context of constructing a quadratic refinement of the intersection pairing on ordinary differential cohomology
in (Freed 14, section 5.1.1) in the context of invertible extended topological field theories.
The Anderson dual of KU is (complex conjugation-equivariantly) the 4-fold suspension spectrum $\Sigma^4 KU$ (Heard-Stojanoska 14, theorem 8.2). This implies that, nonequivariantly $KU$ is Anderson self-dual and the Anderson dual of $KO$ is $\Sigma^4KO$, which were both first proven by Anderson.
Similarly Tmf$[1/2]$ is Anderson dual to its 21-fold suspension (Stojanoska 12).
Original articles include
Donald W. Anderson, Universal coefficient theorems for K-theory, mimeographed notes, Univ. California, Berkeley, Calif., 1969 (pdf)
Zen-ichi Yosimura, Universal coefficient sequences for cohomology theories of CW-spectra, Osaka J. Math. 12 (1975), no. 2, 305–323. MR 52 #9212
See also
The Anderson dual of the sphere spectrum is discussed (in a context of extended TQFTs) in
Michael Hopkins, Isadore Singer, appendix B of, Quadratic Functions in Geometry, Topology, and M-Theory, 2005
Dan Freed, section 5.1.1 of Short-range entanglement and invertible field theories (arXiv:1406.7278)
Daniel Freed, Michael Hopkins, section 5.3 of Reflection positivity and invertible topological phases (arXiv:1604.06527)
The Anderson duals of KU and of tmf are discussed in
Vesna Stojanoska, Duality for Topological Modular Forms, Doc. Math. 17 (2012) 271-311 (arXiv:1105.3968)
Drew Heard, Vesna Stojanoska, K-theory, reality, and duality (arXiv:1401.2581)
In the context of heterotic string theory:
Anderson duality in equivariant stable homotopy theory is discussed in
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