$E_n$-space and n-connected pointed space

Question

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

Answer 1

Denis and ``Ring Spectra'', thanks for the references. I did not treat non-connected spaces in "The geometry of iterated loop spaces", which is why you couldn't find that there. It should have been treated in the immediate sequel "$E_{\infty}$ spaces, group completions, and permutative categories", but that perversely and for no good mathematical reason restricts to the case $n=\infty$. The proof works the same way in general. Max, $n$-connected means $\pi_i = 0$ for $i\leq n$ (you can look it up on Wikipedia if you do not believe me; this is implicit in Ring Spectra's answer).

Briefly, the essential point is to start with an $E_n$-space $X$ and construct from it an $n-1$-connected space $Y_n(X)$ and a natural map $X\longrightarrow \Omega^n Y_n(X)$ that is a weak equivalence if $X$ is connected and a group completion in general, hence a weak equivalence if $\pi_0(X)$ is a group. Conversely, if $Y$ is an $n-1$-connected space, then of course $X=\Omega^n Y$ is a grouplike $E_n$-space. With the construction of Geo, $Y_n(X) = B(\Sigma^n, C_n, X)$ where $C_n$ is the monad associated to an $E_n$-operad. The group completion property for $n\geq 2$ follows from the group completion property for the natural map $\alpha_n\colon C_nX \longrightarrow \Omega^n \Sigma^n X$ (which I didn't yet know how to prove when I wrote Geo, but did thanks to work of Fred Cohen when I wrote the cited sequel). The case $n=1 is classical and special.

Answer 2

When n=∞, this states that the homotopy category of infinite loop spaces is equivalent to the homotopy category of connective spectra; see The geometry of iterated loop spaces by May.

We can also prove this statement in the general case. Let X be a (pointed) n-connected space; then the loop space Ωⁿ⁺¹X is an E_n+1-space, and this is easily checked to be grouplike. So if nConnTop and A_n are the categories of n-connective spaces and E_n+1-spaces, respectively, then the (n+1)-fold loop space functor Ωⁿ⁺¹ induces an equivalence of categories between nConnTop and A_n. (I believe the statement for E_n+1-spaces in general should be somewhere in May's book; I can't find it, though.)

One implication of this, using the Freudenthal suspension theorem, is that if a homotopy n-type has the structure of a grouplike E_n+2-space, then it canonically has the structure of an E_∞-space, which is a special case of the Baez-Dolan stablization hypothesis. (See here, for example.)

RE your edit: a connective spectrum is usually defined as an object of the homotopy limit of the sequence ...→Top_n+1→Top_n→...→Top₁. (See also Denis's comment on your question.)