A¹ homotopy theory

In algebraic geometry and algebraic topology, branches of mathematics, homotopy theory or motivic 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 , which is not an algebraic variety, with the affine line , which is. The theory has seen 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.
 homotopy theory is founded on a category called the homotopy category formula_1. Simply put, the homotopy category, or rather the canonical functor formula_2, is the universal functor from the category formula_3 of smooth formula_4-schemes towards an infinity category which satisfies Nisnevich descent, such that the affine line becomes contractible. Here formula_4 is some prechosen base scheme (e.g., the spectrum of the complex numbers formula_6).
This definition in terms of a universal property is not possible without infinity categories. These were not available in the 90's and the original definition passes by way of Quillen's theory of model categories. Another way of seeing the situation is that Morel-Voevodsky's original definition produces a concrete model for (the homotopy category of) the infinity category formula_1.
This more concrete construction is sketched below.
Step 0.
Choose a base scheme formula_4. Classically, formula_4 is asked to be Noetherian, but many modern authors such as Marc Hoyois work with quasi-compact quasi-separated base schemes. In any case, many important results are only known over a perfect base field, such as the complex numbers, it's perfectly fine to consider only this case.
Step 1.
"Step 1a: Nisnevich sheaves". Classically, the construction begins with the category formula_10 of Nisnevich sheaves on the category formula_3 of smooth schemes over formula_4. Heuristically, this should be considered as (and in a precise technical sense is) the universal enlargement of formula_3 obtained by adjoining all colimits and forcing Nisnevich descent to be satisfied.
"Step 1b: simplicial sheaves". In order to more easily perform standard homotopy theoretic procedures such as homotopy colimits and homotopy limits, formula_14 replaced with the following category of simplicial sheaves.
Let be the simplex category, that is, the category whose objects are the sets
and whose morphisms are order-preserving functions. We let formula_15 denote the category of functors formula_16. That is, formula_15 is the category of simplicial objects on formula_10. Such an object is also called a "simplicial sheaf" on formula_3.
"Step 1c: fibre functors". For any smooth formula_4-scheme formula_21, any point formula_22, and any sheaf formula_23, let's write formula_24 for the stalk of the restriction formula_25 of formula_23 to the small Nisnevich site of formula_21. Explicitly, formula_28 where the colimit is over factorisations formula_29 of the canonical inclusion formula_30 via an étale morphism formula_31. The collection formula_32 is a conservative family of fibre functors for formula_10.
"Step 1d: the closed model structure". We will define a closed model structure on formula_15 in terms of fibre functors. Let formula_35 be a morphism of simplicial sheaves. We say that:
The homotopy category of this model structure is denoted formula_37.
Step 2.
This model structure has Nisnevich descent, but it does not contract the affine line. A simplicial sheaf formula_38 is called formula_39-local if for any simplicial sheaf formula_40 the map
induced by formula_42 is a bijection. Here we are considering formula_39 as a sheaf via the Yoneda embedding, and the constant simplicial object functor formula_44.
A morphism formula_35 is an formula_39-weak equivalence if for any formula_39-local formula_48, the induced map
is a bijection. The formula_39-local model structure is the localisation of the above model with respect to formula_39-weak equivalences.
Formal Definition.
Finally we may define the homotopy category.
Note that by construction, for any in , there is an isomorphism
in the homotopy category.
Properties of the theory.
Wedge and smash products of simplicial (pre)sheaves.
Because we started with a simplicial model category to construct the formula_58-homotopy category, there are a number of structures inherited from the abstract theory of simplicial models categories. In particular, for formula_59 pointed simplicial sheaves in formula_60 we can form the wedge product as the colimitformula_61and the smash product is defined asformula_62recovering some of the classical constructions in homotopy theory. There is in addition a cone of a simplicial (pre)sheaf and a cone of a morphism, but defining these requires the definition of the simplicial spheres.
Simplicial spheres.
From the fact we start with a simplicial model category, this means there is a cosimplicial functorformula_63defining the simplices in formula_60. Recall the algebraic n-simplex is given by the formula_4-schemeformula_66Embedding these schemes as constant presheaves and sheafifying gives objects in formula_60, which we denote by formula_68. These are the objects in the image of formula_69, i.e. formula_70. Then using abstract simplicial homotopy theory, we get the simplicial spheresformula_71We can then form the cone of a simplicial (pre)sheaf asformula_72and form the cone of a morphism formula_73 as the colimit of the diagramformula_74In addition, the cofiber of formula_75 is simply the suspension formula_76. In the pointed homotopy category there is additionally the suspension functorformula_77 given by formula_78and its right adjointformula_79called the loop space functor.
Remarks.
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 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 -sphere in topology, and those coming from the simplicial sphere (considered as constant simplicial sheaf). This leads to a theory of motivic spheres 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 homotopy theory is at least as complicated as classical homotopy theory.
Motivic analogies.
Eilenberg-Maclane spaces.
For an abelian group formula_80 the formula_81-motivic cohomology of a smooth scheme formula_21 is given by the sheaf hypercohomology groupsformula_83for formula_84. Representing this cohomology is a simplicial abelian sheaf denoted formula_85 corresponding to formula_86 which is considered as an object in the pointed motivic homotopy category formula_87. Then, for a smooth scheme formula_21 we have the equivalenceformula_89showing these sheaves represent motivic Eilenberg-Maclane spacespg 3.
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 Gm to become invertible. This process can be carried out either using model-categorical constructions using so-called Gm-spectra or alternatively using infinity-categories.
For "S" = Spec (R), the spectrum of the field of real numbers, there is a functor
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
to an equivalence, since formula_92 is homotopy equivalent to a two-point set. has shown that the resulting functor
is an equivalence.