  We show that loop groups and the universal cover of $\mathrm{Diff}_+(S^1)$
can be expressed as colimits of groups of loops/diffeomorphisms supported in
subintervals of $S^1$. Analogous results hold for based loop groups and for the
based diffeomorphism group of $S^1$. These results continue to hold for the
corresponding centrally extended groups.
  We use the above results to construct a comparison functor from the
representations of a loop group conformal net to the representations of the
corresponding affine Lie algebra. We also establish an equivalence of
categories between solitonic representations of the loop group conformal net,
and locally normal representations of the based loop group.
