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Concept
A¹ homotopy theory
Summary
In algebraic geometry and algebraic topology, branches of mathematics, A1 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 [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 of mixed motives and the proof of the Milnor and Bloch-Kato conjectures.
A1 homotopy theory is founded on a category called the A1 homotopy category . Simply put, the A1 homotopy category, or rather the canonical functor , is the universal functor from the category of smooth -schemes towards an which satisfies Nisnevich descent, such that the affine line A1 becomes contractible. Here 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 . 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 .
This more concrete construction is sketched below.
Choose a base scheme . Classically, 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 1a: Nisnevich sheaves. Classically, the construction begins with the category of Nisnevich sheaves on the category of smooth schemes over . 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.
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