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

Fusion Category Symmetry I: Anomaly In-Flow and Gapped Phases