nLab
A1-homotopy theory

Idea

The ${𝔸}^1$-homotopy category is an analog of the ordinary homotopy category $\mathrm{Ho}(\mathrm{Top})$ 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.

Related concepts

• B1-homotopy theory

References

• Fabien Morel, An introduction to ${𝔸}^1$ homotopy theory, ICTP Trieste July 2002 (directory, pdf, ps).

• Fabien Morel, Vladimir Voevodsky, ${𝔸}^1$-homotopy theory of schemes K-theory, 0305 (web pdf)

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)