  We present a systematic recipe for generating and classifying duality
transformations in one-dimensional quantum lattice systems. Our construction
emphasizes the role of global symmetries, including those described by
(non)-abelian groups but also more general categorical symmetries. These
symmetries can be realized as matrix product operators which allow the
extraction of a fusion category that characterizes the algebra of all symmetric
operators commuting with the symmetry. Known as the bond algebra, its explicit
realizations are classified by module categories over the fusion category. A
duality is then defined by a pair of distinct module categories giving rise to
dual realizations of the bond algebra, as well as dual Hamiltonians. Symmetries
of dual models are in general distinct but satisfy a categorical Morita
equivalence. A key novelty of our categorical approach is the explicit
construction of matrix product operators that intertwine dual bond algebra
realizations at the level of the Hilbert space, and in general map local order
operators to non-local string-order operators. We illustrate this approach for
known dualities such as Kramers-Wannier, Jordan-Wigner, Kennedy-Tasaki and the
IRF-vertex correspondence, a new duality of the $t$-$J_z$ chain model, and
dualities in models with the exotic Haagerup symmetry. Finally, we comment on
generalizations to higher dimensions.
