# 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…

## 3 Citations

### Labelled cospan categories and properads

- Mathematics
- 2022

We prove Steinebrunner’s conjecture on the biequivalence between (colored) properads and labelled cospan categories. The main part of the work is to establish a 1-categorical, strict version of the…

### Fibrations and Koszul duality in locally Cartesian localisations

- Mathematics
- 2021

I show that any locally Cartesian left localisation of a presentable ∞-category admits a right proper model structure in which all morphisms are cofibrations, and obtain a Koszul duality…

### Proper Orbifold Cohomology

- Mathematics
- 2020

The concept of orbifolds should unify differential geometry with equivariant homotopy theory, so that orbifold cohomology should unify differential cohomology with proper equivariant cohomology…

## References

SHOWING 1-10 OF 22 REFERENCES

### SHIFTED COISOTROPIC CORRESPONDENCES

- MathematicsJournal of the Institute of Mathematics of Jussieu
- 2020

Abstract We define (iterated) coisotropic correspondences between derived Poisson stacks, and construct symmetric monoidal higher categories of derived Poisson stacks, where the $i$ -morphisms are…

### Homotopy theory for (braided) cat-groups

- Mathematics
- 1997

Les categories de Gr-categories (tressees ou symetriques) ou les categories equivalentes de modules croises (2-modules croises reduits ou modules stables) representent des modeles algebriques pour…

### Equivariant maps which are self homotopy equivalences

- Mathematics
- 1980

The aim of this note is (i) to give (in §2) a precise statement and proof of the (to some extent well-known) fact mat the most elementary homotopy theory of "simplicial sets on which a fixed…

### NOTES ON 2-GROUPOIDS, 2-GROUPS AND CROSSED MODULES

- Mathematics
- 2005

This paper contains some basic results on 2-groupoids, with special emphasis on computing derived mapping 2-groupoids between 2-groupoids and proving their invariance under strictification. Some of…

### The geometry of iterated loop spaces

- Mathematics
- 1972

Operads and -spaces.- Operads and monads.- A? and E? operads.- The little cubes operads .- Iterated loop spaces and the .- The approximation theorem.- Cofibrations and quasi-fibrations.- The smash…

### Higher Topos Theory

- Philosophy, Mathematics
- 2009

This purpose of this book is twofold: to provide a general introduction to higher category theory (using the formalism of "quasicategories" or "weak Kan complexes"), and to apply this theory to the…

### Cohomology with coefficients in symmetric cat-groups. An extension of Eilenberg–MacLane's classification theorem

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 1993

Abstract In this paper we use Takeuchy–Ulbrich's cohomology of complexes of categories with abelian group structure to introduce a cohomology theory for simplicial sets, or topological spaces, with…