  In this paper we discuss gauging noninvertible zero-form symmetries in two
dimensions. We specialize to certain gaugeable cases, specifically, fusion
categories of the form Rep(H) for H a suitable Hopf algebra (which includes the
special case Rep(G) for G a finite group). We also specialize to the case that
the fusion category is multiplicity-free. We discuss how to construct a
modular-invariant partition function from a choice of Frobenius algebra
structure on H^*. We discuss how ordinary G orbifolds for finite groups G are a
special case of the construction, corresponding to the fusion category Vec(G) =
Rep( C[G]^* ). For the cases Rep(S_3), Rep(D_4), and Rep(Q_8), we construct the
crossing kernels for general intertwiner maps. We explicitly compute partition
functions in the examples of Rep(S_3), Rep(D_4), Rep(Q_8), and Rep(H_8), and
discuss applications in c=1 CFTs. We also discuss decomposition in the special
case that the entire noninvertible symmetry group acts trivially.
