  We develop a mathematical theory of symmetry protected trivial (SPT) orders
and anomaly-free symmetry enriched topological (SET) orders in all dimensions
via two different approaches with an emphasis on the second approach. The first
approach is to gauge the symmetry in the same dimension by adding topological
excitations as it was done in the 2d case, in which the gauging process is
mathematically described by the minimal modular extensions of unitary braided
fusion 1-categories. This 2d result immediately generalizes to all dimensions
except in 1d, which is treated with special care. The second approach is to use
the 1-dimensional higher bulk of the SPT/SET order and the boundary-bulk
relation. This approach also leads us to a precise mathematical description and
a classification of SPT/SET orders in all dimensions. The equivalence of these
two approaches, together with known physical results, provides us with many
precise mathematical predictions.
