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
Cartier duality is a refinement of Pontryagin duality form topological groups to group schemes.
Let $G$ be a finite group scheme over $k$, regarded as a sheaf of groups $G \in Sh(Ring^{op}_k)$. Write $\mathbb{G}_m$ for the multiplicative group, similarly regarded.
Then the Cartier dual $\widehat G$ is the internal hom
of group homomorphisms, hence the sheaf which to $R \in Ring_k^{op}$ assigns the set
of group homomorphisms over $Spec(R)$
This appears for instance as (Polishuk, (10.1.11)).
Cartier duality is indeed a duality in that for any finite commutative group scheme $G$ there is an isomorphism
of the double Cartier dual with the original group scheme.
(e.g. Polishuk, right above (10.1.11), Hida 00, theorem 1.7.1)
A textbook account is for instance in section 10.1 of
or section 1.7 of
lecture notes include
Generalization beyond finite group schemes is discussed in
and in
Discussion in the context of higher algebra (brave new algebra) is in