  The theory of the shift current is thus far geometrical without being
topological. This means that the real-space displacement/shift of a
photoexcited quasiparticle depends on the geometric Berry phase, but the Berry
phase is not quantized to a rational multiple of $2\pi$. I rectify this status
quo by introducing a new class of topological insulators whose band topology is
\textit{only} compatible with a non-centrosymmetric space group. For such
insulators, it is impossible to continuously tune the
$\boldsymbol{k}$-dependent shift vector to zero throughout the Brillouin zone.
Suitably averaged, the shift vector is quantized to a rational multiple of a
Bravais lattice vector. Even with wide band gaps, the frequency-integrated
shift conductivity greatly exceeds $e^3/h^2$, and is at least three orders of
magnitude larger than the conductivity of the prototypical ferroelectric
BaTiO$_3$. The large conductivity is attributed to an interplay between
quantized intra- and inter-band Berry phases. In particular, topological
defects of the inter-band Berry phase can enhance the shift current, even for
unpolarized insulators with negligible intra-band Berry phase.
