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12:44 AM
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.
 
 
12 hours later…
1:06 PM
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)
 
 
2 hours later…
2:53 PM
@AchimKrause Cool! So in particular it's true for Lie groups, which gives another consistency check for me. Thanks!
 

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