  Fusion category symmetries are finite symmetries in 1+1 dimensions described
by unitary fusion categories. We classify 1+1d time-reversal invariant bosonic
symmetry protected topological (SPT) phases with fusion category symmetry by
using topological field theories. We first formulate two-dimensional unoriented
topological field theories whose symmetry splits into time-reversal symmetry
and fusion category symmetry. We then solve them to show that SPT phases are
classified 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 of
orientation-reversing symmetry that is compatible with the fiber functor $(Z,
M, i)$. We apply this classification to SPT phases with Kramers-Wannier-like
self-duality.
