A¹ homotopy theory

Construction

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

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 Δ^opShv(T ) denote the category of functors Δ^op → Shv(T ). That is, Δ^opShv(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 Δ^opShv(T ) in terms of points. Let f : X → 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 ∗ X → x ∗ 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 H s ( T ) .

Step 2

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

The morphism i₀ ∐ i₁ : pt ∐ pt → I is a monomorphism.

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

Hom H s ( T ) ( Y × I , X ) → Hom H s ( T ) ( Y , X )

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

Hom H s ( T ) ( Y , Z ) → Hom H s ( T ) ( X , Z )

is a bijection. The homotopy theory of the site with interval (T, I ) is the localization of Δ^opShv(T ) with respect to I-weak equivalences. This category is called H ( T , I ) .

Formal Definition

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 Δ^opShv_Nis(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
S ≅ X,

in the homotopy category.

Properties of the theory

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 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.