# 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 ðŸ™‚ 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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