Is it true that the homotopy category of group-like $E_n$-spaces is equivalent to the homotopy category of pointed $n$-connected spaces ? If it is true, what should be the statement when $"n\rightarrow \infty"$ ?
By $n$-connected space $X$, I mean that $\pi_{i}X=0$ for $0\leq i\leq n-1$.
Edit
Notions: The $\infty$-category of group-like $E_n$-spaces is denoted by $\mathbf{G}_{n}$ The category of pointed $n$-connected spaces is denoted by $\mathbf{Top}_{n}$. As Peter May and Ring Spectra noticed, $$Bar^{n}:\mathbf{G}_{n}\longrightarrow \mathbf{n-Conn}:\Omega^{n}$$$$Bar^{n}:\mathbf{G}_{n}\longrightarrow \mathbf{Top}_{n}:\Omega^{n}$$ is an $\infty$-equivalence. It seems very natural that the homotpy limit $$ holim(\dots \rightarrow \mathbf{G}_{n+1}\rightarrow \mathbf{G}_{n}\rightarrow\dots \mathbf{G}_{1})$$ is the $\infty$-category of group-like $E_{\infty}$-spaces i.e. connective spectra. My question is the following:
How can we see that
$$ holim(\dots \rightarrow \mathbf{Top}_{n+1}\rightarrow \mathbf{Top}_{n}\rightarrow\dots \mathbf{Top}_{1})$$ is naturally equivalent to the $\infty$-category of connective spectra without using $E_{n}-spaces$?
PS: As Peter May noticed there is a problem with the my definition of $n$-connectivity. But I think the idea is clear.
