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Koszul Duality in Higher Topoi

@article{Beardsley2019KoszulDI,
  title={Koszul Duality in Higher Topoi},
  author={Jonathan Beardsley and Maximilien P'eroux},
  journal={arXiv: Algebraic Topology},
  year={2019}
}
We show that for any pointed and $k$-connective object $X$ of an $n$-topos $\mathcal{X}$ for $0\leq n\leq\infty$ and $k>0$, there is an equivalence between the $\infty$-category of modules in $\mathcal{X}$ over the associative algebra $\Omega^{k} X$, and the $\infty$-category of comodules in $\mathcal{X}$ for the cocommutative coalgebra $\Omega^{k-1}X$. Along the way, we also show that Lurie's straightening-unstraightening equivalence holds over an $(n-1)$-groupoid in any $n$-topos for $0\leq n… 
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