Classifying spaces of E_1 - spaces Hello,
I try to understand aspects of homotopy coherence, in particular "recognition principle" of May.
About the following I did not think a lot, but I decided to ask here anyway, so to save reinvention of the wheel and get clarifying comments.
Consider $E_1$ - the topological operad of small $1$-cubes. An $E_1$-space for me is a space together with an action of $E_1$. I think of an $E_1$-space as of a space together with a multiplication, associative up to "coherent homotopies".
Questions:
1) What will be the definition of a torsor for an $E_1$-space over some base, i.e. the analog of a principal homogeneous space for a topological group.
2) What will be the definition of a classifying space of a particular $E_1$-space.
3) If our $E_1$-space is the loop space $\Omega X$ of some space $X$ (with the satndard $E_1$-action), is true then that $X$ is the classifying space of $\Omega X$.
Probably in the above I did not insert some technical issues involving perhaps words like "group-like" or "fibrant", which I will be happy to hear about.
Thank you,
Sasha
 A: From the horse's mouth.


*

*I would think a good theory of parametrized $E_1$-spaces should not be too
hard to develop, along the general lines of parametrized spaces (and spectra) 
as developed ad nauseum in


J.P. May and J. Sigurdsson. Parametrized homotopy theory.  
Presumably the fibers should be grouplike.  A current student, John Lind, could 
answer better. He is working on classification theorems in a more sophisticated
context of parametrized spectra. 


*

*There are several constructions. My original machine in Geo (The Geometry of
iterated loop spaces), Thm 13.1, gave $B(\Sigma,E_1,X)$ as a delooping of an 
$E_1$-space $X$, using your notation.  (The cited result works for $E_n$-spaces
for all $n$. One can also convert $X$ to an equivalent topological monoid $B(M,E_1,X)$,
by Thm 13.4 of Geo, and take the ordinary classifying space of that. These two constructions 
are compared in papers by Thomason and Fiedorowicz, circa 1980, or maybe earlier.

*This is answered affirmatively for all $n$ in my original work, in part (vi) of
Thm 13.1: $B(\Sigma^n,E_n,\Omega^nY)$ is weakly equivalent to $Y$ if $Y$ is $n$-connected.
The proviso can be improved to $n-1$-connected. It is then obviously necessary, since applying $\Omega^n$ loses any information about $\pi_0$ through $\pi_{n-1}$.
