A¹ homotopy theory

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In algebraic geometry and algebraic topology, a branch 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.

[edit] Construction of the A1 homotopy category

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 several 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 simplicial category, that is, the category whose objects are the sets {0}, {0, 1}, {0, 1, 2}, and so on, 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.

A point of a site T is a geometric morphism x* : Shv(T) → Set, where Set is the category of sets. We will define a closed model structure on ΔopShv(T) in terms of points. Let f : \mathcal{X} \to \mathcal{Y} be a morphism of simplicial sheaves. We say that:

  • f is a weak equivalence if, for any point x of T, the morphism of simplicial sets x^*f : x^*\mathcal{X} \to x^*\mathcal{Y} is a weak equivalence.
  • f is a cofibration if it is a monomorphism.
  • f is a fibration if it has the right lifting property with respect to any cofibration which is a weak equivalence.

The homotopy category of this model structure is denoted \mathcal{H}_s(T).

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:

  • If p is the canonical morphism I → pt, then
    • μ(i0 × 1I) = μ(1I × i0) = i0p.
    • μ(i1 × 1I) = μ(1I × i1) = 1I.
  • The morphism i_0 \amalg i_1 : \text{pt} \amalg \text{pt} \to I is a monomorphism.

Now we localize the homotopy theory with respect to I. A simplicial sheaf \mathcal{X} is called I-local if for any simplicial sheaf \mathcal{Y} the map

\text{Hom}_{\mathcal{H}_s(T)}(\mathcal{Y} \times Y, \mathcal{X}) \to \text{Hom}_{\mathcal{H}_s(T)}(\mathcal{Y}, \mathcal{X})

induced by i0 : pt → I is a bijection. A morphism f : \mathcal{X} \to \mathcal{Y} is an I-weak equivalence if for any I-local \mathcal{Z}, the induced map

\text{Hom}_{\mathcal{H}_s(T)}(\mathcal{Y}, \mathcal{Z}) \to \text{Hom}_{\mathcal{H}_s(T)}(\mathcal{X}, \mathcal{Z})

is a bijection. The homotopy theory of the site with interval (T, I) is the localization of ΔopShv(T) with respect to I-weak equivalences. This category is called \mathcal{H}(T, I).

Finally we may define the A1 homotopy category. Let S be a finite dimensional Noetherian scheme, and let Sm/S denote the category of smooth schemes over S. Equip Sm/S with the Nisnevich topology to get the site (Sm/S)Nis. We let the affine line A1 play the role of the interval. The above construction determines a closed model structure on ΔopShvNis(Sm / S), and the corresponding homotopy category is called the A1 homotopy category.

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