  In this paper, we study the twisted gauging on the (1+1)d lattice and
construct various non-local mappings on the lattice operators. To be specific,
we define the twisted Gauss law operator and implement the twisted gauging of
the finite group on the lattice motivated by the orbifolding procedure in the
conformal field theory, which involves the data of non-trivial element in the
second cohomology group of the gauge group. We show the twisted gauging is
equivalent to the two-step procedure of first applying the SPT entangler and
then untwisted gauging. We use the twisted gauging to construct the triality
(order 3) and $p$-ality (order $p$) mapping on the $\mathbb{Z}_p\times
\mathbb{Z}_p$ symmetric Hamiltonians, where $p$ is a prime. Such novel
non-local mappings generalize Kramers-Wannier duality and they preserve the
locality of symmetric operators but map charged operators to non-local ones. We
further construct quantum process to realize these non-local mappings and
analyze the induced mappings on the phase diagrams. For theories that are
invariant under these non-local mappings, they admit the corresponding
non-invertible symmetries. The non-invertible symmetry will constrain the
theory at the multicritical point between the gapped phases. We further give
the condition when the non-invertible symmetry can have symmetric gapped phase
with a unique ground state.
