Let $Ho(Spc)$ be the homotopy category of spaces. There is an adjoint pair $$ \Sigma^k \colon Ho(Spc) \leftrightarrows Ho(Spc)\colon \Omega^k, $$ where $\Sigma^k$ is the $k$-th supension functor and $\Omega^k$ is the $k$-fold loop space functor ($k\ge 1)$. This adjunction has as associated monad the functor $\Omega^k\Sigma^k$. My question is what are the algebras over this monad (a precise reference to this fact would be welcome).

If we denote by $Ho^s$ the homotopy category of spectra, then there is another adjunction $$ \Sigma^{\infty} \colon Ho(Spc) \leftrightarrows Ho^s\colon \Omega^{\infty}, $$ where $\Sigma^{\infty}$ is the suspension spectrum functor. My question is again what are the algebras over the monad $\Omega^{\infty}\Sigma^{\infty}$.