  Symmetry protected topological (SPT) phases of bosons in $d$ spatial
dimensions have been characterized by the action of the protecting global
symmetry $G$ on their boundary. The symmetry acts on the boundary in a way that
would be impossible to realize in a purely $d-1$ dimensional system i.e.,
without the bulk. This is often formalized by saying the $G$ symmetry is
anomalous when the boundary theory is gauged. Simultaneously gauging the
symmetry on the boundary and in the bulk yields a gauge-invariant composite
system. One says there is an anomaly in-flow from the boundary to the bulk.
Recently it has been appreciated that some anomalies are too severe to be
regulated by an SPT bulk. These do not satisfy anomaly in-flow in the
traditional sense. However, we show that these anomalous systems can be
regulated as the symmetric boundaries of the newly discovered higher SPT
phases. These higher SPT's are protected by a symmetry $d$-group, a higher
categorical version of a group which can charge not just particles but also
strings, volumes, etc. This structure emerges naturally from computation of the
anomaly. One can interpret these severe anomalies as producing an emergent
higher-form gauge field when $G$ is gauged, and this gauge field participates
in the bulk topological action.
