  It is well known that (1+1)-D bosonic symmetry-protected topological (SPT)
phases with symmetry group $G$ can be identified by the projective
representation of the symmetry at the edge. Here, we generalize this result to
higher dimensions. We assume that the representation of the symmetry on the
spatial edge of a ($d+1$)-D SPT is /local/ but not necessarily /on-site/, such
that there is an obstruction to its implementation on a region with boundary.
We show that such obstructions are classified by the cohomology group
$H^{d+1}(G, U(1))$, in agreement with the classification of bosonic SPT phases
proposed in [Chen et al, Science 338, 1604 (2012)]. Our analysis allows for a
straightforward calculation of the element of $H^{d+1}(G, U(1))$ corresponding
to physically meaningful models such as non-linear sigma models with a theta
term in the action. SPT phases outside the classification of Chen et al are
those in which the symmetry cannot be represented locally on the edge. With
some modifications, our framework can also be applied to fermionic systems in
(2+1)-D.
