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 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.

Step 1[edit]

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

{0}, {0, 1}, {0, 1, 2}, ...,

and whose morphisms are order-preserving functions. We let ΔopShv(T ) denote the category of functors ΔopShv(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 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 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 .

Step 2[edit]

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 × II 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 i0i1 : pt ∐ pt → I is a monomorphism.

Now we localize the homotopy theory with respect to I. A simplicial sheaf is called I-local if for any simplicial sheaf the map

induced by i0 : pt → I is a bijection. A morphism is an I-weak equivalence if for any I-local , the induced map

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 .

Formal Definition[edit]

Finally we may define the A1 homotopy category.

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

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

X ×S A1

in the homotopy category.

Properties of the theory[edit]

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 yield 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 Sp,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.