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 $0\leq i\leq n-1$.