(This is a reading seminar given by the PhD student Taeyoon Woo.) In this reading seminar, I will go through the construction of the un/stable motivic homotopy categories and their basic properties. A brief review of the topological side will help an overview. Nisnevich topology and simplicial homotopy theory of sheaves will be the main notions for presenting an ∞-topos of motivic spaces. The unstable motivic homotopy is then defined as A^1-localization, which is modeled by a Bousfield localization. Here I will sketch a proof of the purity theorem after some basic properties. If possible, I will discuss stabilization and the representability of algebraic K-theory.

(This is a reading seminar given by the PhD student Taeyoon Woo.) In this reading seminar, I will go through the construction of the un/stable motivic homotopy categories and their basic properties. A brief review of the topological side will help an overview. Nisnevich topology and simplicial homotopy theory of sheaves will be the main notions for presenting an ∞-topos of motivic spaces. The unstable motivic homotopy is then defined as A^1-localization, which is modeled by a Bousfield localization. Here I will sketch a proof of the purity theorem after some basic properties. If possible, I will discuss stabilization and the representability of algebraic K-theory.