# 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

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From the horse's mouth.

1. 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.

1. 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.

2. 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 $\pi_{n-1}$.

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Thank you very much for your answer. I am reading your book, but I still didn't figure things out. Let me try again to ask my question 2: As I understand, your construction $B(\cdot,E_1,X)$ is a bar construction which gives a way to CONSTRUCT the classifying space. What I hoped for is a DEFINITION of the classifying space, by universal property. Maybe it is not hard, but I don't know homotopy theory well. So could it be something like "a space s.t. a map into it, up to homotopy, is the same as a family...", or maybe "the homotopy type of the space of maps into it is the same as..." –  Sasha Mar 16 '11 at 8:07
If $X$ is grouplike, a classifying space $BX$ should be characterized by an $H$-map $X\to \Omega BX$ that is a weak equivalence. In the $E_2$-case, where one doesn't need to worry about non-commutativity, a two-fold classifying space $B^2X$ should be characterized by an $H$-map $X\to \Omega^2B^2X$ that is a group completion. I didn't study the non-connected case in Geo, but rectified that a bit later in $E_{\infty}$ spaces, group completions, and permutative categories''. –  Peter May Mar 17 '11 at 3:09