  We define a mapping space for Gray-enriched categories adapted to higher
gauge theory. Our construction differs significantly from the canonical mapping
space of enriched categories in that it is much less rigid. The two essential
ingredients are a path space construction for Gray-categories and a kind of
comonadic resolution of the 1-dimensional structure of a given Gray-category
obtained by lifting the resolution of ordinary categories along the canonical
fibration of GrayCat over Cat.
