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
The fact that for a manifold $X$ of dimension $k$, then for any embedding $\iota \colon X\hookrightarrow \mathbb{R}^n$ (which exists by the Whitney embedding theorem) the Thom space $X^{\nu}$ of the normal bundle $\nu$ behaves like a dual to $X_+$ under smash product of pointed topological spaces.
Under passing to suspension spectra this becomes Atiyah duality in stable homotopy theory.
For ex-spaces see around def. 1.4.2 of
For parametrized spectra see page xyz of