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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 insubintervals of $S^1$. Analogous results hold for based loop groups and for thebased diffeomorphism group of $S^1$. These results continue to hold for thecorresponding centrally extended groups.  We use the above results to construct a comparison functor from therepresentations of a loop group conformal net to the representations of thecorresponding affine Lie algebra. We also establish an equivalence ofcategories between solitonic representations of the loop group conformal net,and locally normal representations of the based loop group.

Loop groups and diffeomorphism groups of the circle as colimits