Homotopy Invariance of Presheaves with Transfers 📚

Seminar on A1-topology, motives and K-theory, December 16, 2021
Andrei Druzhinin (SPbU)
Strict homotopy invariance theorem for presheaves with transfers over an arbitrary field

The strict homotopy invariance is a property of presheaves of abelian groups or complexes or S1-spectra that naturally appear whenever the combination of A1-homotopy invariance property and Nisnevich sheafification and localisation are considered together. The strict homotopy invariance theorems for presheaves with transfers allow to compute the motivic localisation in terms of the transfers structures, and play an important role for the computational results and the structural properties of categories for transfers-based motivic theories such as Voevodsky’s motives and Garkusha-Panin’s framed motives.

We extend the generality of the strict homotopy invariance theorems in the latter studies to the case of an arbitrary base field.

At the same time, we discuss formulations of some partial results on box-homotopy invariance that can be obtained at the present moment by applying in such setting geometric constructions already contained in the proof written for A1-homotopy case.