at.algebraic topology – Homotopy coherent generalization of classifying space theory Answer

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at.algebraic topology – Homotopy coherent generalization of classifying space theory

$\newcommand{\cS}{\mathcal{S}}\newcommand{\Fun}{\mathrm{Fun}}\newcommand{\LMod}{\mathrm{LMod}}\newcommand{\Sp}{\mathrm{ Sp}}$Hey Laurent 🙂 Let $X$ be a space (which I’ll view as a Kan complex), and let $\cS$ denote the $\infty$-category of spaces. You could think of the homotopy-coherent categorification of “maps $X \to BG$ up to homotopy” as $\cS_{/X}$. Then Theorem 2.2.1.2 of Higher Topos Theory (see Section 2.2 of https://arxiv.org/pdf/1403.4325.pdf for a bite-sized summary) says that there’s an equivalence $\Fun(X, \cS) \simeq \cS_{/X}$. So, take $X = BG$ (ie, the nerve of the one-object topological category whose morphism space is $G$), we see that $\Fun(BG, \cS) \simeq \cS_{/BG}$. But a functor from $BG \to \cS$ is exactly the data of a homotopy-coherent $G$– action on a space, as desired. (One other useful fact is that if $X$ is assumed to be connected, and $\Sp$ denotes the $\infty$-category of spectra, then there’s a Koszul duality equivalence $\Fun(X, \Sp) \simeq \LMod_{\Sigma^\infty_+ \Omega X}(\Sp)$where $\Sigma^\infty_+ \Omega X$ is viewed as an $\mathbf{E}_1$-algebra in $\Sp$. Taking $X = BG$we see that $\Fun(BG, \Sp) \simeq \LMod_{\Sigma^\infty_+ G}(\Sp)$; you can interpret this as saying that the $\mathbf{E}_1$-algebra $\Sigma^\infty_+ G$ plays the role of a spherical group ring of $G$.)

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