  A key aspect of a recent proposal for a {\em generalized loop representation}
of quantum Yang-Mills theory and gravity is considered. Such a representation
of the quantum theory has been expected to arise via consideration of a
particular algebra of observables -- given by the traces of the holonomies of
{\em generalized loops}. We notice, however, a technical subtlety, which
prevents us from reaching the conclusion that the generalized holonomies are
covariant with respect to small gauge transformations. Further analysis is
given which shows that they are {\em not} covariant with respect to small gauge
transformations; their traces are {\em not} observables of the gauge theory.
This result indicates what may be a serious complication to the use of
generalized loops in physics.
