## Thesis: Motivic homotopy theory in derived algebraic geometry

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*Last update:* 2018-09-11 (corrected some typos)

*Abstract:*
In topology, generalized cohomology theories are representable in the stable homotopy category.
The analogue in algebraic geometry is the stable motivic homotopy category, constructed by Morel–Voevodsky, where generalized motivic cohomology theories are representable.
In this thesis, we consider an extension of this construction to the world of derived algebraic geometry.
We demonstrate the Morel–Voevodsky localization theorem in this setting, and establish the formalism of Grothendieck’s six operations.
This formalism is encoded together with all homotopy coherence data using the language of higher category theory.

*Contents:*

Chapter 0: *Preliminaries*

Review of the theories of (∞,1)-categories and derived algebraic geometry.

Chapter 1: *Motivic spaces and spectra*

Construction of the categories and basic functorialities, proof of the localization theorem.

Chapter 2: *The formalism of six operations*

Construction of the six functor formalism, lifting SH to the (∞,2)-category of correspondences.