  We study generalized discrete symmetries of quantum field theories in 1+1D
generated by topological defect lines with no inverse. In particular, we
describe 't Hooft anomalies and classify gapped phases stabilized by these
symmetries, including new 1+1D topological phases. The algebra of these
operators is not a group but rather is described by their fusion ring and
crossing relations, captured algebraically as a fusion category. Such data
defines a Turaev-Viro/Levin-Wen model in 2+1D, while a 1+1D system with this
fusion category acting as a global symmetry defines a boundary condition. This
is akin to gauging a discrete global symmetry at the boundary of
Dijkgraaf-Witten theory. We describe how to "ungauge" the fusion category
symmetry in these boundary conditions and separate the symmetry-preserving
phases from the symmetry-breaking ones. For Tambara-Yamagami categories and
their generalizations, which are associated with Kramers-Wannier-like
self-dualities under orbifolding, we develop gauge theoretic techniques which
simplify the analysis. We include some examples of CFTs with fusion category
symmetry derived from Kramers-Wannier-like dualities as an appetizer for the
Part II companion paper.
