Scheiderer motives and equivariant higher topos theory

We give an algebro-geometric interpretation of $C_2$-equivariant stable homotopy theory by means of the $b$-topology introduced by Claus Scheiderer in his study of $2$-torsion phenomena in \'etale cohomology. To accomplish this, we first revisit and extend work of Scheiderer on equivariant topos theory by functorially associating to a $\infty$-topos $\mathscr{X}$ with $G$-action a presentable stable $\infty$-category $\mathrm{Sp}^G(\mathscr{X})$, which recovers the $\infty$-category $\mathrm{Sp}^G$ of genuine $G$-spectra when $\mathscr{X}$ is the terminal $G$-$\infty$-topos. Given a scheme $X$ with $1/2 \in \mathcal{O}_X$, our construction then specializes to produce an $\infty$-category $\mathrm{Sp}^{C_2}_b(X)$ of "$b$-sheaves with transfers" as $b$-sheaves of spectra on the small \'etale site of $X$ equipped with certain transfers along the extension $X[i] \rightarrow X$; if $X$ is the spectrum of a real closed field, then $\mathrm{Sp}^{C_2}_b(X)$ recovers $\mathrm{Sp}^{C_2}$. On a large class of schemes, we prove that, after $p$-completion, our construction assembles into a premotivic functor satisfying the full six functors formalism. We then introduce the $b$-variant $\mathrm{SH}_b(X)$ of the $\infty$-category $\mathrm{SH}(X)$ of motivic spectra over $X$ (in the sense of Morel-Voevodsky), and produce a natural equivalence of $\infty$-categories $\mathrm{SH}_b(X)^{\wedge}_p \simeq \mathrm{Sp}^{C_2}_b(X)^{\wedge}_p$ through amalgamating the \'etale and real \'etale motivic rigidity theorems of Tom Bachmann. This involves a purely algebro-geometric construction of the $C_2$-Tate construction, which may be of independent interest. Finally, as applications, we deduce a "$b$-rigidity" theorem, use the Segal conjecture to show \'etale descent of the $2$-complete $b$-motivic sphere spectrum, and construct a parametrized version of the $C_2$-Betti realization functor of Heller-Ormsby.