Comonadicity of spaces over spectra?

Question

As connective spectra are equivalent to group-like $E_{\infty}$ algebras in spaces, the $\infty$-category of connective spectra is monadic over the $\infty$-category of spaces though the usual $\Sigma^{\infty} \dashv \Omega^{\infty}$ adjunction.

Is it also the case that the category of spaces is comonadic over the category of spectra (or maybe connective spectra) ?

If so do we have a nice description of "what is a (unstable) space" in terms of stable homotopy theory, i.e. can we say something about what are the $\Sigma^{\infty} \Omega^{\infty}$-coalgebra in spectra in terms of more familiar structure (in the same way that a $\Omega^{\infty} \Sigma^{\infty}$-algebra is the same as a group-like $E_{\infty}$-algebra).

For example $\Sigma^{\infty} X$ naturally has the structure of a cocomutative (in the $E_{\infty}$-sense) co-algebras (coming from the diagonal map of $X$). Can we characterize $\Sigma^{\infty} \Omega^{\infty}$-coalgebra, as such cocommutative coalgebra satisfying some additional conditions ?

I'm interested in how to deduce some properties of a higher toposes from properties of its category of spectra, and more generally, how much of a higher topos can be understood from its category spectra. so this sort of explicit description could be useful and any description of the co-monad only involving more classical structure on the category of spectra (like the smash product and the subcategory of connective spectra) might be useful.

Answer 1

Let me record this as an answer: Blomquist and Harper show that $\Sigma^\infty$ is comonadic on simply-connected spaces.

It occurs to me that if your ultimate goal is to understand unstable homotopy theory in terms of stable homotopy theory, you may also be interested in the following monadicity result in chromatic homotopy theory: For each $n$, the $v_n$-periodic localization of pointed, $d$-connected spaces (where $d = d(n)$ is some fixed number which grows with $n$) is monadic over the category of $v_n$-periodic spectra, via the Bousfield-Kuhn functor. Moreover, the monad in question is precisely the monad for Lie algebras. That is, the $\infty$-category of pointed, $d$-connected, $v_n$-periodic spaces is equivalent to the category of Lie algebras in the category of $v_n$-periodic spectra. This is almost enough to understand spaces entirely in terms of stable homotopy theory, in that $v_n$-periodic homotopy groups detect weak equivalences (as $n$ varies) between simply-connected finite spaces.

However, there's the gap in that the category of Lie algebras corresponds to a category of pointed, $d(n)$-connected spaces. So even understanding all of this Lie algebra data won't tell you anything about a space which depends only on finitely many homotopy groups. And since it's a result about pointed spaces it won't be straightforward to interpret topos-theoretically anyway.