Given a site equipped with an interval object the homotopy localization of an (∞,1)-category of (∞,1)-sheaves on is the (∞,1)-categorical localization of at the morphisms of the form
Taking Top and the interval object to be the standard topological interval , the homotopy localization of -stacks on is equivalent to the (∞,1)-category Top itself again. For more on this see the discussion and references at topological ∞-groupoid.
Taking the category of relative schemes over a Noetherian scheme and taking the affine line, the study of the corresponding homotopy localization is called A1 homotopy theory.
A homotopy localization of the (∞,1)-topos of ∞-stacks on the Nisnevich site is used in motivic cohomology. See there for more details.