Title: Stratification and duality for homotopical groups,
Abstract: In this paper, we show that the category of module spectra over
$C^*(B\mathcal{G},\mathbb{F}_p)$ is stratified for any $p$-local compact group
$\mathcal{G}$, thereby giving a support-theoretic classification of all
localizing subcategories of this category. To this end, we generalize Quillen's
$F$-isomorphism theorem, Quillen's stratification theorem, Chouinard's theorem,
and the finite generation of cohomology rings from finite groups to homotopical
groups. Moreover, we show that $p$-compact groups admit a homotopical form of
Gorenstein duality.