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
Instances of “dualities” relating two different, maybe opposing, but to some extent equivalent concepts or phenomena are ubiquitous in mathematics (and in mathematical physics, see at dualities in physics).
In terms of general abstract concepts in category theory instances of dualities might be (and have been) organized as follows:
involution – any automorphism which is an involution, hence which squares to the identity? may be though of as exhibiting two dual perspectives on the objects that it acts on;
abstract duality – the operation of sending a category to its opposite category is such an involution on Cat itself (and in fact this is the only non-trivial automorphism of Cat, see here). This has been called abstract duality. While the construction is a priori tautologous, any given opposite category often is equivalent to a category known by other means, which makes abstract duality interesting.
concrete duality – given a closed category $\mathcal{C}$ and any object $D$ of it, then the operation $[-,D] : \mathcal{C} \to \mathcal{C}^{op}$ obtained by forming the internal hom into $D$ sends each object to something like a $D$-dual object. This is particularly so if $D$ is indeed a dualizing object in a closed category in that applying this operation twice yields an equivalence of categories $[[-,D],D] : \mathcal{C} \stackrel{\simeq}{\to} \mathcal{C}$ (so that $[-,D]$ is a (contravariant) involution on $\mathcal{C}$). If $\mathcal{C}$ is in addition a closed monoidal category then under some conditions on $D$ (but not in general) this kind of concrete dualization coincides with the concept of forming dual objects in monoidal categories.
From (Lawvere-Rosebrugh, chapter 7):
Not every statement will be taken into its formal dual by the process of dualizing with respect to $V$, and indeed a large part of the study of mathematics
space vs. quantity
and of logic
theory vs. example
may be considered as the detailed study of the extent to which formal duality and concrete duality into a favorite $V$ correspond or fail to correspond. (p. 122)
adjunction – another categorical concept of duality is that of adjunction, as in pairs of adjoint functors. Via the many incarnations of universal constructions in category theory this accounts for all dualities that arise as instances as the dual pairs
left and right Kan extension
(Given that the saying has it that “Everything in mathematics is a Kan extension”, this goes some way in explaining the ubiquity of duality in mathematics.)
When the adjoint functors are monads and hence modalities, then adjointness between them has been argued to specifically express the concept of duality of opposites.
Adjunctions and specifically dual adjunctions (“Galois connections”) may be thought of as a generalized version of the above abstract duality: every dual adjunction induces a maximal dual equivalence between subcategories.
dual object, dualizing object, dualizing object in a closed category
the duality between space and quantity
Poincaré duality for finite dimensional (oriented) closed manifolds
Spanier-Whitehead duality, Brown-Comenetz duality, Anderson duality
Pontryagin duality for commutative (Hausdorff) topological groups
Cartier duality of a finite flat commutative group scheme
Serre duality on nonsingular projective algebraic varieties which has as a special case the statement of the Riemann-Roch theorem
Grothendieck duality, coherent duality? for coherent sheaves?
Verdier duality for abelian categories of sheaves; e.g. for a category of sheaves of abelian groups.
Artin-Verdier duality? generalizing Tate duality? for constructible sheaves over the spectrum of a ring of algebraic numbers
Of particular interest are concrete dualities between concrete categories $C, D$, i.e. categories equipped with faithful functors
to Set, which are represented by objects $a \in C$, $\hat a \in D$ with the same underlying set $f(a) = \hat f(\hat a)$. Such objects are known as dualizing objects.
Lawvere and Rosebrugh, chaper 7 of Sets for Mathematics (web)
H.-E. Porst, W. Tholen, Concrete Dualities in Category Theory at Work, Herrlich, Porst (eds.) pdf
David Corfield; More on duality (blog)
wikipedia duality (mathematics)
MathOverflow: the-concept-of-duality
Discussion of duality specifically in homological algebra and stable homotopy theory with emphasis on the concept of dualizing object in a closed category (and the induced Umkehr maps etc.) is in