  A well-defined local non-Abelian gauge connection involving a rank-p gauge
B-field was introduced a decade ago. This was achieved by introducing doublet
groups and doublet-assembled connections that can act on a doublet of matter
fields, and so, standard gauging procedures and minimal substitutions are
applicable.
  We provide here a summarized version of this formalism. We also build up
doublet-extended gauge-invariant actions for bosonic and fermionic matter
fields, and discuss the appearance of novel topological quantities in these
doublet-type gauge models. A partner action for higher-spin fields appears in
the doublet version of the fermionic matter sector. As an application of the
proposed formalism, a gauge-invariant Chern-Simons action in four dimensions is
built up; a second order quadratic Yang-Mills type action can also be defined
in this doublet framework, but we show that they can only be invariant under a
Lie subgroup of our extended gauge symmetry. We finally carry out the task of
studying the topological mass generation mechanism in the simplest model
combining these actions into a non-Abelian generalization of the
Cremmer-Scherk-Kalb-Ramond theory that we also show to be power-counting
renormalizable.
