  This paper derives a large web of exact lattice dualities in one and two
spatial dimensions. Some of the dualities are well-known, while others, such as
two-dimensional boson-parafermion dualities, are new. The procedure is
systematic, independent of specific Hamiltonians, and generalizes to higher
dimensions. One important result is a demonstration that spin structures in
arbitrary lattice fermion theories can always be simply defined as topological
gauge fields whose gauge group is the fermion number parity. This definition
agrees with other expected properties of spin structures, and it motivates the
introduction of "paraspin structures" that serve the same role in parafermion
theories.
