Geodesic
Koszul duality in higher topoi
Homology, homotopy, and applications, Tome 25 (2023) no. 1, pp. 53-70.

Voir la notice de l'article provenant de la source International Press of Boston

We show that there is an equivalence in any $n$-topos $\mathscr{X}$ between the pointed and $k$-connective objects of $\mathscr{X}$ and the $\mathbb{E}_k$-group objects of the $(n-k-1)$-truncation of $\mathscr{X}$. This recovers, up to equivalence of $\infty$-categories, some classical results regarding algebraic models for $k$-connective, $(n-1)$-coconnective homotopy types. Further, it extends those results to the case of sheaves of such homotopy types. We also show that for any pointed and $k$-connective object $X$ of $\mathscr{X}$ there is an equivalence between the $\infty$-category of modules in $\mathscr{X}$ over the associative algebra $\Omega^k X$, and the $\infty$-category of comodules in $\mathscr{X}$ for the cocommutative coalgebra $\Omega^{k-1} X$. All of these equivalences are given by truncations of Lurie’s $\infty$-categorical bar and cobar constructions, hence the terminology “Koszul duality.”
DOI : 10.4310/HHA.2023.v25.n1.a3
Classification : 16T15, 18D35, 55U30 
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Jonathan Beardsley; Maximilien Péroux. Koszul duality in higher topoi. Homology, homotopy, and applications, Tome 25 (2023) no. 1, pp. 53-70. doi : 10.4310/HHA.2023.v25.n1.a3. http://geodesic.mathdoc.fr/articles/10.4310/HHA.2023.v25.n1.a3/

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