We will start with a quick introduction to motivic homotopy theory, emphasizing the parallels to the homotopy theory of topological spaces.

We will then develop some basic motivic homotopy theory in an abstract setup: We replace the infinity category of motivic spaces by a presentable, cartesian closed infinity category (PCCC) and the multiplicative group scheme by a commutative group object G in this category. Starting from this, we will construct basic geometric objects like projective spaces, see a representation theorem for G-bundles, a Snaith type algebraic K-theory spectrum, Adams operations, rational splittings and a rational Eilenberg-MacLane spectrum.

Examples of our abstract setup include motivic spaces built from complex and non-archimedian analytic geometry, derived geometry, log geometry as well as the many proposals for geometry over the field with one element. Furthermore, there is an initial example of our setup, the classifying PCCC for commutative groups, which is a model for univalent homotopy type theory. Results and constructions carried out in this particular PCCC with its generic commutative group carry over to general PCCCs with commutative groups and could be modeled in type theory.