The term *duality* can refer to many notions. Most of them are instances of the notion of [[opposite category|dual category]]. Often a duality is formulated in terms of [[cohomology]]. * [[opposite category|dual category]] * [[dual vector space]] * [[Poincaré duality]] for finite dimensional (oriented) closed manifolds * [[Pontryagin duality]] for commutative ([[Hausdorff space|Hausdorff]]) [[topological group|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 [[Rieman-Roch theorem]] * [[Grothendieck duality]], [[coherent duality]] for [[coherentsheave|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 number|algebraic numbers]].