  Dualities play an important role in mathematics and they can be described with the help of equivalences between categories. In other words, many important mathematical theorems can be translated as statements about the existence of adjoint functors, sometimes satisfying additional properties. This is sometimes taken as expressing the conceptual content of the theorem. Consider the following fundamental case: let \(\mathbf{C}\) be the category whose objects are the locally compact abelian groups and the morphisms are the continuous group homomorphisms. Then, the Pontryagin duality theorem amounts to the claim that the category \(\mathbf{C}\) is equivalent to the category \(\mathbf{C}\)°, that is, to the opposite category. Of course, the precise statement requires that we describe the functors \(F: \mathbf{C} \rightarrow \mathbf{C}\)° and \(G: \mathbf{C}\)° \(\rightarrow \mathbf{C}\) and prove that they constitute an equivalence of categories.