Motivic homotopy theory

Summary

Motivic homotopy theory, also called $\mathbb{A}^1$-homotopy theory, is a homotopy theory for smooth schemes. In it the affine line $\mathbb{A}^1$ plays the role of the unit interval. It was founded by Morel and Voevodsky. Algebraic K-theory, motivic cohomology and algebraic cobordism become representable objects in it. Voevodsky used it in his proof of the Milnor conjecture.