Profinite Borel completeness and smooth Artin motives

Abstract

The purpose of this paper is twofold. In the first part, we revisit the description of the $\infty$-category of Borel complete equivariant spectra for a finite group given by Mathew-Naumann-Noel, introduce a version with coefficients, and then consider Borel equivariance for profinite groups. Here we identify two generally differing notions: levelwise Borel completeness and the hypercompletion thereof. In the second part, we study variants of smooth Artin motives, which are subcategories of the $\infty$-categories of effective Nisnevich and \'etale Voevodsky motives over a base scheme $S$ that are controlled by the \'etale fundamental group $\pi_1^{\mathrm{\'et}}(S)$. In the Nisnevich case, we extend a theorem of Voevodsky and identify smooth Artin motives with modules over the Bredon cohomology spectrum for the profinite group $\pi_1^{\mathrm{\'et}}(S)$. In the \'etale case, we show that the difference between our two notions of profinite Borel completeness is precisely the difference between \'etale sheaves and hypersheaves on finite \'etale schemes.