Motivic Homotopy Theory

Throughout the talk, $S$ is a quasi-compact and quasi-separated scheme.

Definition 0.1

Let $\mathrm{Sm}_S$ be the category of finitely presented smooth schemes over $S$ .

Since we impose the finitely presented condition, this is an essentially small (1-)category.

The starting point of motivic homotopy theory is the $\infty$ -category $\mathcal{P}(\mathrm{Sm}_S)$ of presheaves on $\mathrm{Sm}_S$ . This is a symmetric monoidal category under the Cartesian product. Eventually, we will need the pointed version $\mathcal{P}(\mathrm{Sm}_S)_*$ , which can be defined either as the category of pointed objects in $\mathcal{P}(\mathrm{Sm}_S)$ , or the category of presheaves with values in pointed spaces. This is symmetric monoidal $\infty$ -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:

$\begin{useimager} \[ \begin{tikzcd} \Pre(\Sm_S) \ar[r, "L_{\Nis}"] \ar[d, "\widetilde{L_{\A^1}}"] & L_{\Nis} \Pre(\Sm_S) \equiv \Spc_S \ar[d, "L_{\A^1}"]\\ L_{\A^1} \Pre(\Sm_S) \ar[r, "\widetilde{L_{\Nis}}"] & L_{A^1 \wedge \Nis} \Pre(\Sm_S) \equiv \Spc_S^{\A^1} \equiv \H(S) \end{tikzcd} \] \end{useimager}$

The first chapter will discuss the horizontal arrows (i.e. Nisnevich localization), and the second will discuss the vertical ones (i.e. $\mathbb {A}^1$ -localization).

Afterwards, we will start “doing homotopy theory”. In chapter 3, we will discuss the motivic version of homotopy groups, and in chapter 4, we will discuss Thom spaces.

In chapter 5, we will introduce the stable motivic category, and finally, in chapter 6, we will introduce effective and very effective spectra.