# Contents

## Idea

The ${𝔸}^{1}$-homotopy category is an analog of the ordinary homotopy category $\mathrm{Ho}\left(\mathrm{Top}\right)$ of the category Top with topological spaces replaced by ∞-stacks over certain schemes.

It is a special case of the homotopy theory induced form any sufficiently well-behaved interval object $I$ in a site $C$ on the ∞-stacks on $C$, for the case that $C$ is the category of smooth schemes over a Noetherian scheme and

$I={𝔸}^{1}$I = \mathbb{A}^1

is the standard affine line in $C$.

## Examples and applications

• One example of ${𝔸}^{1}$-homotopy theory appears in motivic cohomology. See there for more details.

• ${𝔸}^{1}$-homotopy theory sheds light on (and was at least partially motivated by) the proof of the Bloch-Kato conjecture?.

• There is an analog of ${𝔸}^{1}$-homotopy theory for other geometries. The extra left adjoint on an cohesive (infinity,1)-topos may realize the localization at an abstract continuum line object. See at cohesion for more details.

## References

For more on the general procedure see homotopy localization.

Discussion related to étale homotopy is in

• Daniel Isaksen, Étale realization of the ${𝔸}^{1}$-homotopy theory of schemes, Advances in Mathematics 184 (2004)

Revised on June 13, 2013 01:40:16 by Urs Schreiber (131.174.43.123)