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".
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