A¹ homotopy theory

A¹ homotopy theory

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.

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 Δop"Shv"("T") denote the category of functors Δop → "Shv"("T"). That is, Δop"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") → "Set", where "Set" is the category of sets. We will define a closed model structure on Δop"Shv"("T") in terms of points. Let f : mathcal{X} o 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} o 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("i"0 × 1"I") = μ(1"I" × "i"0) = "i"0"p".
** μ("i"1 × 1"I") = μ(1"I" × "i"1) = 1"I".
* The morphism i_0 amalg i_1 : ext{pt} amalg ext{pt} o 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

: ext{Hom}_{mathcal{H}_s(T)}(mathcal{Y} imes Y, mathcal{X}) o ext{Hom}_{mathcal{H}_s(T)}(mathcal{Y}, mathcal{X})

induced by "i"0 : "pt" → "I" is a bijection. A morphism f : mathcal{X} o mathcal{Y} is an "I"-weak equivalence if for any "I"-local mathcal{Z}, the induced map

: ext{Hom}_{mathcal{H}_s(T)}(mathcal{Y}, mathcal{Z}) o ext{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 Δop"Shv"("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 Delta^ ext{op} ext{Shv}_ ext{Nis}( ext{Sm}/S), and the corresponding homotopy category is called the A1 homotopy category.

References

* | year=1999 | journal=Publications Mathématiques de l'IHÉS | issn=1618-1913 | issue=90 | pages=45–143


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