Motivic model categories and motivic derived algebraic geometry

TL;DR

This paper develops an advanced framework called motivic derived algebraic geometry, extending derived algebraic geometry to $A^1$-homotopy theory, and constructs key geometric and cohomological objects within this setting.

Contribution

It introduces motivic versions of $ty$-categories, topoi, schemes, and stacks, and constructs algebraic $K$-theory, Grassmannians, Thom spaces, and algebraic cobordism in this new framework.

Findings

Constructed motivic algebraic $K$-theory and Grassmannians.

Formulated motivic Thom spaces and algebraic cobordism.

Proved algebraic cobordism corepresents the universal motivic $ty$-category.

Abstract

In this paper, we develop an enhancement of derived algebraic geometry to apply to $\mathbb{A}^1$-homotopy theory introduced by Morel and Voevodsky. We call the enhancement "motivic derived algebraic geometry". We shall actually formulate "motivic" versions of $\infty$-categories, $\infty$-topoi, spectral schemes and spectral Deligne--Mumford stacks established by Joyal, Lurie, To\"en and Vezzosi. By using the language of motivic derived algebraic geometry, we construct the Grassmannian and the algebraic $K$-theory. Furthermore we formulate the Thom spaces for vector bundles on (motivic) stacks, and we obtain the algebraic cobordism for (motivic) stacks. As the main result, we prove that the algebraic cobordism corepresents the motivic $\infty$-category which has the universal property of oriented (motivic) $\infty$-categories.