By Fabien Morel

^{1}-homotopy idea over a base box, i.e., with the traditional homotopy conception linked to the class of tender kinds over a box within which the affine line is imposed to be contractible. it's a typical sequel to the foundational paper on A

^{1}-homotopy idea written including V. Voevodsky. encouraged by way of classical ends up in algebraic topology, we current new strategies, new effects and purposes relating to the homes and computations of A

^{1}-homotopy sheaves, A

^{1}-homology sheaves, and sheaves with generalized transfers, in addition to to algebraic vector bundles over affine soft varieties.

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