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 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.
Simply put, the A1 homotopy category, or rather the canonical functor
-schemes towards an infinity category which satisfies Nisnevich descent, such that the affine line A1 becomes contractible.
is some prechosen base scheme (e.g., the spectrum of the complex numbers
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.
is asked to be Noetherian, but many modern authors such as Marc Hoyois work with quasi-compact quasi-separated base schemes.
In any event, many important results are only known over a perfect base field, such as the complex numbers, so we consider only this case.
Heuristically, this should be considered as (and in a precise technical sense is) the universal enlargement of
obtained by adjoining all colimits and forcing Nisnevich descent to be satisfied.
We say that: The homotopy category of this model structure is denoted
This model structure has Nisnevich descent, but it does not contract the affine line.
as a sheaf via the Yoneda embedding, and the constant simplicial object functor
Because we started with a simplicial model category to construct the
recovering 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.
From the fact we start with a simplicial model category, this means there is a cosimplicial functor
Embedding these schemes as constant presheaves and sheafifying gives objects in
In the pointed homotopy category there is additionally the suspension functor
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.
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).
Representing this cohomology is a simplicial abelian sheaf denoted
which is considered as an object in the pointed motivic homotopy category
showing these sheaves represent motivic Eilenberg-Maclane spaces[1]pg 3.
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