Koszul duality in higher topoi (Q6039221)

Classical Koszul duality, sometimes called \ bar-cobar duality, gives an equivalence between suitably derived versions of algerbas and coalgebras in a symmetric monoidal category rigged out in some version of homotopy theory. It was shown in [\textit{J. P. May}, The geometry of iterated loop spaces. Berlin-Heidelberg-New York: Springer-Verlag (1972; Zbl 0244.55009)] that every grouplike \(\mathcal{A}_{\infty}\)-algebra in spaces is equivalent to the space of loops on a pointed and connected space. This is extended to a Quillen equivalence of model categories between simplicial groups and \textit{reduced} simplicial sets [\textit{P. G. Goerss} and \textit{J. F. Jardine}, Simplicial homotopy theory. Basel: Birkhäuser (1999; Zbl 0949.55001)]. This paper generalizes this type of result in two different directions, first showing that such a result holds in an arbitrary \(n\)-topos, and then demonstrating that this categorifies to an equivalence of \(\infty\)-categories between modules over an algebra and comodules over the Koszul dual of coalgebras. This relationship between modules and comodules is already established in the literature for the category of spaces. For an arbitrary space \(X\), the duality between left \(\Omega X\)-modules and \(X\)-comodules in spaces is given in [\textit{E. Dror} et al., Proc. Am. Math. Soc. 80, 670--672 (1980; Zbl 0454.55018); \textit{M. A. Shulman}, Topology Appl. 155, No. 5, 412--432 (2008; Zbl 1196.55014)]. A pointed version of this equivalence is given in [\textit{K. Hess} and \textit{B. Shipley}, Adv. Math. 290, 1079--1137 (2016; Zbl 1331.19003), Theorem 4.14]. Following \textit{J. Lurie} [Higher topos theory. Princeton, NJ: Princeton University Press (2009; Zbl 1175.18001); \url{https://www.math.ias.edu/~lurie/papers/HA.pdf}], this paper exploits the \(\infty\)-category of \(\infty \)-groupoids, denoted \(\mathcal{S}\), to model topological spaces or simplicial sets. Theorem 2.5 recalls the equivalence between \(k\)-connective objects of an \(\infty\)-topos and \(\mathbb{E}_{k}\)-group objects of the same \(\infty\)-topos, which is situated as a Koszul duality result, being rephrased as a relationship between algebras and coalgebras (Theorem 2.26). The main result is Theorem 3.1 extending the equivalence between algebras and coalgebras to one between categories of modules and comodules over those algebras and coalgebras in any \(n\)-topos.