  We show that the theory of MV-algebras is Morita-equivalent to that of
abelian $\ell$-groups with strong unit. This generalizes the well-known
equivalence between the categories of set-based models of the two theories
established by D. Mundici in 1986, and allows to transfer properties and
results across them by using the methods of topos theory. We discuss several
applications, including a sheaf-theoretic version of Mundici's equivalence and
a bijective correspondence between the geometric theory extensions of the two
theories.
