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 groupe 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 cocomutative 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.

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.