  Generalized global symmetries, in particular non-invertible and categorical
symmetries, have become a focal point in the recent study of quantum field
theory (QFT). In this paper, we investigate aspects of symmetry topological
field theories (SymTFTs) and anomalies of non-invertible symmetries for 2D QFTs
from a string theory perspective. Our primary focus is on an infinite class of
2D QFTs engineered on D1-branes probing toric Calabi-Yau 4-fold singularities.
We derive 3D SymTFTs from the topological sector of IIB supergravity and
discuss the resulting 2D QFTs, which can be intrinsically relative or absolute.
For intrinsically relative QFTs, we propose a sufficient condition for them to
exist. For absolute QFTs, we show that they exhibit non-invertible symmetries
with an elegant brane origin. Furthermore, we find that these non-invertible
symmetries can suffer from anomalies, which we discuss from a top-down
perspective. Explicit examples are provided, including theories for
$Y^{(p,k)}(\mathbb{P}^2)$, $Y^{(2,0)}(\mathbb{P}^1\times \mathbb{P}^1)$, and
$\mathbb{C}^4/\mathbb{Z}_4$ geometries.
