A¹ homotopy theory

In algebraic geometry and algebraic topology, a branch of mathematics, $A 1$ homotopy theory is a way to apply the techniques of algebraic topology, specifically homotopy, to algebraic varieties and, more generally, to schemes. The theory is due to Fabien Morel and Vladimir Voevodsky. The underlying idea is that it should be possible to develop a purely algebraic approach to homotopy theory by replacing the unit interval $[0, 1]$ , which is not an algebraic variety, with the affine line $A 1$ , which is. The theory requires a substantial amount of technique to set up, but has spectacular applications such as Voevodsky's construction of the derived category of mixed motives and the proof of the Milnor and Bloch-Kato conjectures.

$A 1$ homotopy theory is founded on a category called the $A 1$ homotopy category. This is the homotopy category for a certain closed model category whose construction requires two steps.

Most of the construction works for any site $T$ . Assume that the site is subcanonical, and let $Shv (T)$ be the category of sheaves of sets on this site. This category is too restrictive, so we will need to enlarge it. Let $Δ$ be the simplex category, that is, the category whose objects are the sets

and whose morphisms are order-preserving functions. We let $Δ op Shv (T)$ denote the category of functors $Δ op \to Shv (T)$ . That is, $Δ op Shv (T)$ is the category of simplicial objects on $Shv (T)$ . Such an object is also called a simplicial sheaf on $T$ . The category of all simplicial sheaves on $T$ is a Grothendieck topos.

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