Homotopy Theory Seminar

Friday, May 6, 2022 - 2:00pm

Peter Haine

UC-Berkeley

Wickelgren and Williams used Morel’s unstable A1-connectivity Theorem to deduce that the James Splitting holds for A1-connected motivic spaces over a perfect field. In this talk we’ll explain why a more fundamental splitting holds for motivic spaces over an arbitrary base scheme. In fact, a modern take on a classical proof of the James Splitting shows that a this more fundamental splitting holds in any ∞-category where all of the relevant terms are defined and pushout squares remain pushouts after basechange along an arbitrary morphism. This gives new contexts for this splitting, including profinite homotopy theory and (equivariant) motivic homotopy theory. We’ll explain why some other classical splitting results hold at this level of generality and give some new descriptions of motivic spaces constructed from P1 ∖ {0,1,∞} . This is joint work with Sanath Devalapurkar.