A¹ homotopy theory

In algebraic geometry and algebraic topology, branches 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.

Construction

$A 1$ 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.

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 $Δ op Shv (T)$ 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.

A point of a site $T$ is a geometric morphism $x * : Shv (T) \to Set$ , where $Set$ is the category of sets. We will define a closed model structure on $Δ op Shv (T)$ 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 ${\mathcal {H}}_{s}(T)$ .

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 $μ : I \times I \to I$ and two maps $i 0, i 1 : pt \to I$ such that:

If $p$ is the canonical morphism $I \to pt$ , then

μ (i 0 \times 1 I) = μ (1 I \times i 0) = i 0 p .

μ (i 1 \times 1 I) = μ (1 I \times i 1) = 1 I .

The morphism $i 0 ∐ i 1 : pt ∐ pt \to I$ 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 I,{\mathcal {X}})\to {\text{Hom}}_{{\mathcal {H}}_{s}(T)}({\mathcal {Y}},{\mathcal {X}})

induced by $i 0 : pt \to I$ 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 $Δ op Shv (T)$ with respect to $I$ -weak equivalences. This category is called ${\mathcal {H}}(T,I)$ .

Formal Definition

Finally we may define the $A 1$ 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

A 1

play the role of the interval. The above construction determines a closed model structure on

Δ op Shv Nis (Sch / S)

, and the corresponding homotopy category is called the

A 1

homotopy category.

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

X \times S A 1 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 $A 1$ 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 $A 1$ homotopy theory is at least as complicated as classical homotopy theory.

The stable homotopy category

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 G_m to become invertible. This process can be carried out either using model-categorical constructions using so-called G_m-spectra or alternatively using infinity-categories.

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

SH(\mathbf {R} )\to SH

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

\rho :S^{0}\to \mathbf {G} _{m},i.e.,\{-1,1\}\to Spec\mathbf {R} [x,x^{-1}]

to an equivalence, since $\mathbf {R} ^{\times }$ is homotopy equivalent to a two-point set. Bachmann (2018) has shown that the resulting functor

SH(\mathbf {R} )[\rho ^{-1}]\to SH

is an equivalence.

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References

Survey Articles

Antieau, Benjamin; Elmanto, Elden, A primer for unstable motivic homotopy theory, arXiv:1605.00929, Bibcode:2016arXiv160500929A

References

Bachmann, Tom (2018), Motivic and Real Etale Stable Homotopy Theory, arXiv:1608.08855

Morel, Fabien; Voevodsky, Vladimir (1999), "A¹-homotopy theory of schemes" (PDF), Publications Mathématiques de l'IHÉS, 90 (90): 45–143, doi:10.1007/BF02698831, MR 1813224, retrieved 9 May 2008
Voevodsky, Vladimir (1998), "A¹-homotopy theory" (PDF), Documenta Mathematica, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998): 579–604, ISSN 1431-0635, MR 1648048

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