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We extend the standard localization theory for function and section spacesdue to Hilton-Mislin-Roitberg and Moller outside the CW category to the case ofcompact metric domain in the presence of a grouplike structure. We studyapplications in two cases directly generalizing the gauge group of a principalbundle. We prove an identity for the monoid of fibre-homotopy self-equivalencesof a Hurewicz fibration -- due to Gottlieb and Booth-Heath-Morgan-Piccinini inthe CW category -- in the compact case. This leads to an extended localizationresult for this monoid. We also obtain an extended localization theory forgroups of sections of a fibrewise group. We give two applications in rationalhomotopy theory.

Localization of grouplike function and section spaces with compact domain