In algebraic geometry and algebraic topology, branches of mathematics, A1 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 A1, 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.

A1 homotopy theory is founded on a category called the A1 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 ΔopShv(T ) denote the category of functors Δop → Shv(T ). That is, ΔopShv(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.

This model structure will not give the right homotopy category because it does not pay any attention to the unit interval object. Call this object I, and denote the final object of T by pt. We assume that I comes with a map μ : I × I → I and two maps i0, i1 : pt → I such that:

Finally we may define the A1 homotopy category.

Note that by construction, for any X in Sch/S, there is an isomorphism

in the homotopy category.

The setup, especially the Nisnevich topology, is chosen as to make algebraic K-theory representable by a spectrum, and in some aspects to make a proof of the Bloch-Kato conjecture possible.

After the Morel-Voevodsky construction there have been several different approaches to A1 homotopy theory by using other model category structures or by using other sheaves than Nisnevich sheaves (for example, Zariski sheaves or just all presheaves). Each of these constructions yields the same homotopy category.

There are two kinds of spheres in the theory: those coming from the multiplicative group playing the role of the 1-sphere in topology, and those coming from the simplicial sphere (considered as constant simplicial sheaf). This leads to a theory of motivic spheres S p,q with two indices. To compute the homotopy groups of motivic spheres would also yield the classical stable homotopy groups of the spheres, so in this respect A1 homotopy theory is at least as complicated as classical homotopy theory.

A further construction in A1-homotopy theory is the category SH(S), which is obtained from the above unstable category by forcing the smash product with Gm to become invertible. This process can be carried out either using model-categorical constructions using so-called Gm-spectra or alternatively using infinity-categories.

For S = Spec (R), the spectrum of the field of real numbers, there is a functor

to the stable homotopy category from algebraic topology. The functor is characterized by sending a smooth scheme X / R to the real manifold associated to X. This functor has the property that it sends the map

is an equivalence.