@S.carmeli Fair enough. Thanks!

@SaalHardali I have in mind infinite-dimensional Banach spaces more specifically than just infinite-dimensional manifolds -- after all a topos has distinguished objects like the initial and terminal objects, which behave kind of like 0 in a Banach space. It also has operations like pullback and pushout which behave kind of like addition in a Banach space (maybe also multiplication in a Banach algebra). But I do think the "infinite-dimensional" part might be important.

Random chromatic question: when we say "$K(\infty) = H\mathbb F_p$", what do we mean? Is this a statement about Bousfield classes? Like is $\langle H \mathbb F_p \rangle$ a maximal Bousfield class? Are we supposed to go further and think of $H \mathbb F_p$-modules as behaving analogously to $K(n)$-local spectra for $n$ finite?