I will describe the A^1-homotopy category and basic constructions therein. Motivated by geometry and classical homotopy theory, it is natural to attempt to study homotopy of maps between two smooth varieties by simply replacing the unit interval by A^1. I will explain how this ``naive" notion of homotopy is problematic in general, but still a very useful guide in a number of situations of interest. To keep the discussion concrete, I will focus on the study of vector bundles on smooth affine varieties.
Seminar
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