  Topological order of a topological phase of matter in two spacial dimensions
is encoded by a unitary modular (tensor) category (UMC). A group symmetry of
the topological phase induces a group symmetry of its corresponding UMC.
Gauging is a well-known theoretical tool to promote a global symmetry to a
local gauge symmetry. We give a mathematical formulation of gauging in terms of
higher category formalism. Roughly, given a UMC with a symmetry group $G$,
gauging is a 2-step process: first extend the UMC to a $G$-crossed braided
fusion category and then take the equivariantization of the resulting category.
Gauging can tell whether or not two enriched topological phases of matter are
different, and also provides a way to construct new UMCs out of old ones. We
derive a formula for the $H^4$-obstruction, prove some properties of gauging,
and carry out gauging for two concrete examples.
