# Homotopy localisation

## Idea

Given a site $C$ equipped with an interval object ${*}\amalg {*} \stackrel{[i_0, i_1]}{\to}I$ the homotopy localization of an (∞,1)-category of (∞,1)-sheaves $Sh_\infty(C)$ on $C$ is the (∞,1)-categorical localization of $Sh_\infty(C)$ at the morphisms of the form

$X \times I \to X \,.$

## Examples

• Taking $C =$ Top and the interval object $I$ to be the standard topological interval $I = [0,1]$, the homotopy localization of $\infty$-stacks on $Top$ is equivalent to the (∞,1)-category Top itself again. For more on this see the discussion and references at topological ∞-groupoid.

• A homotopy localization of the (∞,1)-topos of ∞-stacks on the Nisnevich site is used in motivic homotopy theory. See there for more details.

Revised on February 26, 2014 16:41:06 by Adeel Khan (108.40.108.250)