Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

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The categories of this paper are: hep-ph
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The categories of this paper are: cs
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The categories of this paper are: acc-phys
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The categories of this paper are: supr-con
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The categories of this paper are: mtrl-th
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The categories of this paper are: alg-geom
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The categories of this paper are: nlin
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The categories of this paper are: nucl-ex
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The categories of this paper are: ao-sci
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The categories of this paper are: chem-ph
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The categories of this paper are: funct-an
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The categories of this paper are: solv-int
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The categories of this paper are: patt-sol
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The categories of this paper are: hep-lat
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The categories of this paper are: physics
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The categories of this paper are: gr-qc