A primer for unstable motivic homotopy theory
  In this expository article, we give the foundations, basic facts, and first examples of unstable motivic homotopy theory with a view towards the approach of Asok-Fasel to the classification of vector bundles on smooth complex affine varieties. Our focus is on making these techniques more accessible to algebraic geometers. 
Benjamin Antieau, Elden Elmanto
math.AG math.KT