Relatedly, I'm wondering whether it's possible to have a commutative, cocommutative Hopf algebra in a symmetric monoidal category whose $p$-power map is trivial for some $p$ and is dualizable but not self-dual. $\Sigma^\infty_+ G$ is dualizable for a compact Lie group $G$, but the only commutative ones are $(S^1)^{\times n}$ and finite $G$.

$(S^1)^{\times n}$ has nontrivial $p$-power maps while finite $G$ are self-dual.

$L_{K(n)} \Sigma^\infty_+ BC_p$ is commutative and has trivial $p$-power map and is dualizable $K(n)$-locally, but it too is self-dual.

Is the notion of geometric embedding (i.e. a fully faithful geometric morphism) of \infty-topoi stable under base change in RTop?

@TimCampion just FYI, this is not true for oriented manifolds, but it IS true for stably framed ones.

(I mean that the top cell splits off)

@AchimKrause Cool! So in particular it's true for Lie groups, which gives another consistency check for me. Thanks!