  One of the main features of covariant theories, in particular general
relativity, is that the field equation possesses gauge freedom associated with
global diffeomorphisms of the underlying manifold. I shall explain here how the
hole argument is a reflection of this gauge freedom.
  Finally I shall point out some implications of the hole argument and extend
the hole argument to the case of permutable theories. As covariant theories
provides a general mathematical framework for classical physics, permutable
theories provide the language for quantum physics. Permutable theories are
defined functorially on the category of sets and permutations with values into
the category of fibered sets and fiber-preserving automorphisms, and rules of
selecting sections. The hole argument for permutable theories is intimately
related with the individuation problem of the base elements (e.g., elementary
particles). This is a consequence of the fact that the automorphisms of the
base space provides the gauge freedom of the fields.
