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Unifying Koszul dualities via point-set models

Dan Petersen, Victor Roca i Lucio, Sinan Yalin

Abstract

The classical bar-cobar adjunction between dg algebras and dg coalgebras goes back to the origins of differential homological algebra as developed by Cartan, Eilenberg, Moore, and many others, and is part of the broader framework of Koszul duality. In recent years, several $\infty$-categorical analogues of this adjunction have been developed, notably by Lurie, Francis--Gaitsgory, and Heuts. However, there is no comparison in the literature between the classical chain-level constructions and their higher-categorical counterparts, and in fact the two constructions are not quite compatible. In this paper we provide a unified framework relating these different forms of Koszul duality in the differential graded setting. We construct a commutative square of adjunctions, called the inclusion-restriction square, intertwining the classical operadic bar-cobar adjunction with its completed variant due to Le Grignou--Lejay. We show that this square induces an $\infty$-categorical adjunction between algebras and their Koszul dual coalgebras, recovering in particular the differential graded case of Lurie's bar-cobar adjunction, and explain precisely how our constructions relate to those of Francis--Gaitsgory and Heuts.

Unifying Koszul dualities via point-set models

Abstract

The classical bar-cobar adjunction between dg algebras and dg coalgebras goes back to the origins of differential homological algebra as developed by Cartan, Eilenberg, Moore, and many others, and is part of the broader framework of Koszul duality. In recent years, several -categorical analogues of this adjunction have been developed, notably by Lurie, Francis--Gaitsgory, and Heuts. However, there is no comparison in the literature between the classical chain-level constructions and their higher-categorical counterparts, and in fact the two constructions are not quite compatible. In this paper we provide a unified framework relating these different forms of Koszul duality in the differential graded setting. We construct a commutative square of adjunctions, called the inclusion-restriction square, intertwining the classical operadic bar-cobar adjunction with its completed variant due to Le Grignou--Lejay. We show that this square induces an -categorical adjunction between algebras and their Koszul dual coalgebras, recovering in particular the differential graded case of Lurie's bar-cobar adjunction, and explain precisely how our constructions relate to those of Francis--Gaitsgory and Heuts.

Paper Structure

This paper contains 61 sections, 52 theorems, 127 equations.

Table of Contents

  1. Bar-cobar duality
  2. The associative case
  3. The inclusion-restriction square
  4. Main results
  5. Francis--Gaitsgory's enhanced bar-cobar adjunction.
  6. A dual version of the enhanced bar-cobar adjunction.
  7. An infinity-categorical inclusion-restriction square
  8. Acknowledgements
  9. Conventions
  10. Recollections on algebras and coalgebras
  11. Point-set preliminaries
  12. Algebras over operads and coalgebras over cooperads.
  13. Algebras over a cooperad
  14. The canonical filtration on algebras over cooperads.
  15. Coalgebras over an operad
  16. ...and 46 more sections

Key Result

Theorem A

Let $\Bbbk$ be a field of any characteristic. Let $\mathsf{dg}~\mathsf{assoc}~\mathsf{alg}$ and $\EuScript{A}_\infty \hbox{-}\mathsf{coalg}$ denote the relative categories of dg associative algebras, and $\EuScript A_\infty$-coalgebras, respectively, with weak equivalences given by quasi-isomorphism

Theorems & Definitions (110)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Theorem E: Theorems \ref{['thm: identification of the infinity monad Q']} and \ref{['thm: point-set models for complete enhanced bar-cobar adjunctions']}
  • Theorem F: Theorem \ref{['thm: infinity categorical inclusion-restriction square']}
  • Definition 1.1: dg $\EuScript{P}$-algebra
  • Definition 1.2: dg $\EuScript{C}$-coalgebra
  • Definition 1.4: dg $\EuScript{C}$-algebra
  • Definition 1.6: Canonical filtration on a dg $\EuScript{C}$-algebra
  • ...and 100 more