  We explore exact generalized symmetries in the standard 2+1d lattice
$\mathbb{Z}_2$ gauge theory coupled to the Ising model, and compare them with
their continuum field theory counterparts. One model has a (non-anomalous)
non-invertible symmetry, and we identify two distinct non-invertible symmetry
protected topological phases. The non-invertible algebra involves a lattice
condensation operator, which creates a toric code ground state from a product
state. Another model has a mixed anomaly between a 1-form symmetry and an
ordinary symmetry. This anomaly enforces a nontrivial transition in the phase
diagram, consistent with the "Higgs=SPT" proposal. Finally, we discuss how the
symmetries and anomalies in these two models are related by gauging, which is a
2+1d version of the Kennedy-Tasaki transformation.
