  We study the structure of soft gluon corrections to multi-leg scattering
amplitudes in a non-Abelian gauge theory by analysing the corresponding product
of semi-infinite Wilson lines. We prove that diagrams exponentiate such that
the colour factors in the exponent are fully connected. This completes the
generalisation of the non-Abelian exponentiation theorem, previously proven in
the case of a Wilson loop, to the case of multiple Wilson lines in arbitrary
representations of the colour group. Our proof is based on the replica trick in
conjunction with a new formalism where multiple emissions from a Wilson line
are described by effective vertices, each having a connected colour factor. The
exponent consists of connected graphs made out of these vertices. We show that
this readily provides a general colour basis for webs. We further discuss the
kinematic combinations that accompany each connected colour factor, and
explicitly catalogue all three-loop examples, as necessary for a direct
computation of the soft anomalous dimension at this order.
