By Fabien Morel

^{1}-homotopy thought over a base box, i.e., with the usual homotopy conception linked to the class of gentle forms over a box within which the affine line is imposed to be contractible. it's a normal sequel to the foundational paper on A

^{1}-homotopy concept written including V. Voevodsky. encouraged by way of classical leads to algebraic topology, we current new innovations, new effects and purposes regarding 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 tender varieties.

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