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,  $$Bar^{n}:\mathbf{G}_{n}\longrightarrow \mathbf{n-Conn}:\Omega^{n}$$
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.