By Fabien Morel

this article offers with A1-homotopy concept over a base box, i.e., with the normal homotopy concept linked to the class of tender forms over a box within which the affine line is imposed to be contractible. it's a ordinary sequel to the foundational paper on A1-homotopy conception written including V. Voevodsky. encouraged by means of classical ends up in algebraic topology, we current new innovations, new effects and purposes on the topic of the homes and computations of A1-homotopy sheaves, A1-homology sheaves, and sheaves with generalized transfers, in addition to to algebraic vector bundles over affine soft varieties.

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