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Fusion category symmetries are finite symmetries in 1+1 dimensions describedby unitary fusion categories. We classify 1+1d time-reversal invariant bosonicsymmetry protected topological (SPT) phases with fusion category symmetry byusing topological field theories. We first formulate two-dimensional unorientedtopological field theories whose symmetry splits into time-reversal symmetryand fusion category symmetry. We then solve them to show that SPT phases areclassified by equivalence classes of quintuples $(Z, M, i, s, \phi)$ where $(Z,M, i)$ is a fiber functor, $s$ is a sign, and $\phi$ is the action oforientation-reversing symmetry that is compatible with the fiber functor $(Z,M, i)$. We apply this classification to SPT phases with Kramers-Wannier-likeself-duality.

Topological field theories and symmetry protected topological phases with fusion category symmetries