Throughout the talk, is a quasi-compact and quasi-separated scheme.
Let be the category of finitely presented smooth schemes over .
The starting point of motivic homotopy theory is the -category of presheaves on . This is a symmetric monoidal category under the Cartesian product. Eventually, we will need the pointed version , which can be defined either as the category of pointed objects in , or the category of presheaves with values in pointed spaces. This is symmetric monoidal -category under the (pointwise) smash product.
In the first two chapters, we will construct the unstable motivic category, which fits in the bottom-right corner of the following diagram:
The first chapter will discuss the horizontal arrows (i.e. Nisnevich localization), and the second will discuss the vertical ones (i.e. -localization).