  Using the numerical modular bootstrap, we constrain the space of 1+1d CFTs
with a finite non-invertible global symmetry described by a fusion category
$\mathcal{C}$. We derive universal and rigorous upper bounds on the lightest
$\mathcal{C}$-preserving scalar local operator for fusion categories such as
the Ising and Fibonacci categories. These numerical bounds constrain the
possible robust gapless phases protected by a non-invertible global symmetry,
which commonly arise from microscopic lattice models such as the anyonic
chains. We also derive bounds on the lightest $\mathcal{C}$-violating local
operator. Our bootstrap equations naturally arise from a "slab construction",
where the CFT is coupled to the 2+1d Turaev-Viro TQFT, also known as the
Symmetry TFT.
