Localization of grouplike function and section spaces with compact domain

We extend the standard localization theory for function and section spaces due to Hilton-Mislin-Roitberg and Moller outside the CW category to the case of compact metric domain in the presence of a grouplike structure. We study applications in two cases directly generalizing the gauge group of a principal bundle. We prove an identity for the monoid of fibre-homotopy self-equivalences of a Hurewicz fibration -- due to Gottlieb and Booth-Heath-Morgan-Piccinini in the CW category -- in the compact case. This leads to an extended localization result for this monoid. We also obtain an extended localization theory for groups of sections of a fibrewise group. We give two applications in rational homotopy theory.