This entry is about a concept of duality in general category theory. For the concept of dualizing objects in a closed category as used in homological algebra and stable homotopy theory see at dualizing object in a closed category.
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
A dualizing object is an object $a$ which can be regarded as being an object of two different categories $A$ and $B$, such that the concrete duality which is induced by homming into that object induces duality adjunctions between $A$ and $B$, schematically:
Many famous dualities are induced this way, for instance Stone duality and Gelfand-Naimark duality.
There are various different terms for “dualizing objects”. As recalled on p. 112 of the article by Porst and Tholen below
Isbell speaks of objects keeping summer and winter homes;
Lawvere speaks of objects sitting in two categories;
Simmons speaks of schizophrenic objects.
It has been convincingly argued by Tom Leinster (blog comment here) that the term “schizophrenic” should not be used. Todd Trimble then suggested the term “ambimorphic object.” Another suggestion was “Janusian object.”
Let $A$ and $B$ be two concrete categories, i.e. categories equipped with faithful functor to Set
Then consider pairs of objects $(a \in A, b \in B)$ with the same underlying set, $U(a) \simeq V(b)$. Then … (see the references Dimov-Tholen below).
Examples appear at
G. D. Dimov, W. Tholen, A Characterization of Representable Dualities, In: Categorical Topology and its Relation to Analysis, Algebra and Combinatorics, Prague, Czechoslovakia 22-27 August 1988, J. Adamek and S. MacLane (eds.), World Scientific, Singapore, New Jersey, London, Hong Kong, 1989, pp. 336-357.
G. D. Dimov, W. Tholen, Groups of Dualities, Trans. Amer. Math. Soc., 336 (2), 901-913, 1993. (pdf)
H.-E. Porst, W. Tholen, Concrete Dualities in Category Theory at Work, Herrlich, Porst (eds.) (pdf)
Michael Barr, John F. Kennison, R. Raphael, Isbell Duality (pdf)