Lisse extensions of weaves
Adeel A. Khan
TL;DR
The paper develops a general framework to transport Grothendieck's six-functor formalism from schemes to Artin stacks via lisse extensions, introducing the notions of weaves and preweaves to encode pullbacks, pushforwards, shrieks, and monoidal structures in a coherent ∞-categorical setting. Central to the approach is the lisse extension, which realizess presheaves on spaces as presheaves on Artin stacks by taking right Kan extensions along smooth-locally-smooth atlases, together with descent conditions (étale/Nisnevich) to ensure a unique, well-behaved extension into higher stacks. The authors establish smooth and proper axioms that guarantee the existence and functoriality of all six operations on Artin stacks, prove Poincaré duality in this context, and show that the lisse-extended theories satisfy descent and base-change compatibilities. They also provide concrete examples across algebraic and topological settings, including étale and Nisnevich descent for algebraic stacks and sheaves on topological stacks, illustrating the broad applicability of the construction to motivic, spectral, and topological contexts. Overall, the work delivers a robust, axiomatized pathway to extend sophisticated sheaf theories to broader geometric frameworks while preserving the core six-functor formalism and duality structures, with implications for motivic homotopy theory, topological stacks, and beyond.
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.