Homotopy localisation

Idea

Given a site $C$ equipped with an interval object $* ⨿ * \overset{[i_{0}, i_{1}]}{\to} I$ the homotopy localization of an (∞,1)-category of (∞,1)-sheaves ${Sh}_{∞} (C)$ on $C$ is the (∞,1)-categorical localization of ${Sh}_{∞} (C)$ at the morphisms of the form

X \overset{Id \times i_{0}}{\to} X \times I .

David Roberts: From what I understand of the work on $A^{1}$ homotopy, the localisation is of morphisms of the form $X \times I \to X$ .

Urs Schreiber: but should that make a difference?

Taking $C =$ Top and the interval object $I$ to be the standard topological interval $I = [0,1]$ , the homotopy localization of $∞$ -stacks on $Top$ is equivalent to the (∞,1)-category Top itself again. For more on this see the discussion and references at topological ∞-groupoid.
Taking $C =$ $Sch / S$ the category of relative schemes over a Noetherian scheme $S$ and taking $I = 𝔸^{1}$ 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.