Classifying spaces of E_1 - spaces - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T09:05:25Z http://mathoverflow.net/feeds/question/58541 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/58541/classifying-spaces-of-e-1-spaces Classifying spaces of E_1 - spaces Sasha 2011-03-15T15:28:51Z 2011-03-15T20:12:04Z <p>Hello,</p> <p>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.</p> <p>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".</p> <p>Questions:</p> <p>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.</p> <p>2) What will be the definition of a classifying space of a particular $E_1$-space.</p> <p>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 $\Omega X$.</p> <p>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.</p> <p>Thank you, Sasha</p> http://mathoverflow.net/questions/58541/classifying-spaces-of-e-1-spaces/58568#58568 Answer by Peter May for Classifying spaces of E_1 - spaces Peter May 2011-03-15T20:12:04Z 2011-03-15T20:12:04Z <p>From the horse's mouth.</p> <ol> <li>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</li> </ol> <p>J.P. May and J. Sigurdsson. Parametrized homotopy theory. </p> <p>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. </p> <ol> <li><p>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.</p></li> <li><p>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 $\pi_{n-1}$.</p></li> </ol>