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Despite a pretty thorough look (I think) I can’t find the answer to the following question: Is every (reasonable?) path connected space weakly equivalent to the classifying space of some topological group?

I’ve found some related results, such as the Kan-Thurston theorem, but nothing that answers this question.

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    $\begingroup$ This is true. If X is a pointed, path-connected space, then its loop space $\Omega X$ is weakly equivalent to a topological group and $X$ itself is weakly equivalent to its classifying space. This is perhaps most easily proven by working with simplicial sets, see Goerss-Jardine V.5 and V.6. $\endgroup$
    – Piotr Pstrągowski
    Commented Jun 23, 2020 at 5:02
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    $\begingroup$ To expand on what Piotr said, there is an equivalence of categories between so-called 'grouplike E_1-spaces' and pointed connected spaces. It is then a classical theorem that every grouplike E_1-space admits a strict model (due to Stasheff I think? Although he would have called an E_1 space an A_∞-space). $\endgroup$
    – Harry Gindi
    Commented Jun 23, 2020 at 8:29
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    $\begingroup$ @PiotrPstrągowski Might as well post your excellent (and highly-upvoted) comment as an answer. $\endgroup$
    – Theo Johnson-Freyd
    Commented Jun 23, 2020 at 13:32

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