Lisse extensions of weaves

TL;DR

This paper demonstrates that the process of lisse extension in sheaf theory preserves the structure of Grothendieck's six operations when extending from schemes to Artin stacks, with broad applicability.

Contribution

It establishes that lisse extension maintains the six functor formalism for sheaves on schemes, extending this to Artin stacks and beyond.

Findings

Lisse extension preserves the six operations formalism.

The framework applies to stable motivic homotopy categories.

Applicable to sheaves of spectra on topological stacks.

Abstract

Any sheaf theory on schemes extends canonically to Artin stacks via a procedure called lisse extension. In this paper we show that lisse extension preserves the formalism of Grothendieck's six operations: more precisely, the lisse extension of a weave on schemes determines a weave on (higher) Artin stacks. The setup is general enough to apply to the stable motivic homotopy category with the six functor formalism of Voevodsky-Ayoub-Cisinski-Deglise, for instance, and is not specific to algebraic geometry: for example, it also applies to sheaves of spectra on topological stacks.