TITLE: Iterated free infinite loop spaces
QUESTION [10 upvotes]: Let $Q$ denote $\Omega^\infty\circ \Sigma^\infty$ the free infinite loop space functor. Given some space $X$, we see that $QX$ carries all the stable homotopy information about $X$. Naturally I wanted to do this again $QQX$, but examples such as spheres and eilenberg-maclane spaces get too complicated.
There is a natural maps $\iota :X \to QX$ which means we can have a tower:
$$X \to QX \to Q^2X \to \cdots$$
Taking the colimit I can imagine there being an object $Q^\infty X$. So my questions are:

What is known about repeated applications of the free infinite loop space functor?
Can it be iterated as I have described? What does this describe?

REPLY [4 votes]: If $X$ is connected then $QX$ is stably equivalent to the total extended power $DX$ (by the Snaith splitting, as others have mentioned), so $QQX=QDX$.  If $X$ is not connected then $QX$ is still stably equivalent to a group completion of $DX$.  So it is useful to think about iterating $D$, and then you can go back to thinking about $Q$ if you want.
A key example to consider is as follows: if $G$ is a groupoid, and $SG$ is the free symmetric monoidal category on $G$, then $\Sigma^\infty BG_+=K(SG)$, and $DBG=BSG$.  If $G$ is just a group (considered as a one-object groupoid) then $SG$ is the groupoid of pairs $(P\xrightarrow{p}X)$, where $P$ and $X$ are finite sets, $G$ acts freely on $P$, and $p$ induces a bijection $P/G\to X$.  In other words, $P$ is a $G$-torsor over $X$. 
 More generally, we have $D^nBG=BS^nG$, where $S^nG$ is the groupoid of diagrams 
$$ (P\xrightarrow{p}X_{1}\xrightarrow{f_1}\dotsb\xrightarrow{}X_n), $$
where all the $X_i$ are finite sets, the maps $f_i$ are arbitrary, and $P$ is a $G$-torsor over $X_1$.  The natural map $D^nBG\to D^{n+1}BG$ corresponds to the functor that just adds $X_{n+1}=\{0\}$ at the end.  So $D^\infty BG=BS^\infty G$, where $S^\infty G$ is the analogous groupoid of diagrams 
$$ (P\xrightarrow{p}X_{1}\xrightarrow{f_1}X_2 \to X_3 \to \dotsb ) $$
where $|X_n|=1$ for $n\gg 0$.