Homotopy localisation

Idea

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

$X×I\to X\phantom{\rule{thinmathspace}{0ex}}.$X \times I \to X \,.

Examples

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

• Taking $C=$ $\mathrm{Sch}/S$ 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 homotopy theory. See there for more details.

Revised on August 8, 2013 18:49:06 by Marc Hoyois (92.104.194.50)