Homology, Homotopy and Applications

Volume 25 (2023)

Number 1

Koszul duality in higher topoi

Pages: 53 – 70

DOI: https://dx.doi.org/10.4310/HHA.2023.v25.n1.a3


Jonathan Beardsley (Department of Mathematics and Statistics, University of Nevada, Reno, Nv., U.S.A.)

Maximilien Péroux (Department of Mathematics, David Rittenhouse Laboratory, University of Pennsylvania, Philadelphia, Penn., U.S.A.)


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.”

2010 Mathematics Subject Classification

16T15, 18D35, 55U30

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Copyright © 2023, Jonathan Beardsley and Maximilien Péroux. Permission to copy for private use granted.

Received 7 February 2021

Received revised 20 December 2021

Accepted 22 December 2021

Published 1 March 2023