Motivic Homotopy Theory and Synthetic Spectra

Abstract

Motivic homotopy studies the application of techniques from homotopy theory to algebraic geometry, using A1 as the analogue of the unit interval. Voevodsky found early success in using his constructions to prove Milnor’s conjecture and the Bloch-Kato conjecture. An interesting and deep theory arises when constructing the unstable and stable motivic categories, and it has developed into its own field of study. We begin with a survey of these constructions, detailing the equivalences between the different model used in the construction of H(S) and SH(S). Here, we draw connections between all the constructions one might encounter across the literature, and provide explicit statements on their equivalence.

Stable homotopy theorists have also found utility in motivic homotopy, using the stable mo- tivic homotopy category SH to advance computations of the stable homotopy groups of spheres, such as in the work of Isaksen-Wang-Xu. Other work by Bachmann-Kong-Wang-Xu has made great progress in our understanding of motivic homotopy theory.

Synthetic spectra are a construction of Pstragrowski which represent a ‘return to form’ ofsome sort, as they are constructed entirely in the ∞−category of spectra. However, they give rise to a natural bigrading and a strong connection to motivic homotopy; one of the main results is an equivalence of ∞−categories with cellular motivic spaces over C, SpCcell. We build up enough of the general theory to establish the connection with motivic homotopy and comment on recent applications.