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Global Koszul Duality: Differential Graded Cocommutative Coalgebras and Curved Lie Algebras

Alexander Mallon, You Wang

Abstract

We give a combinatorial model structure to the category of, not necessarily conilpotent, differential graded (dg) cocommutative coalgebras and an $\infty$-category structure to the category of curved Lie algebras over an algebraically closed field of characteristic $0$. Further, we extend the Harrison and Chevally-Eilenberg functors between dg cocommutative conilpotent coalgebras and dg Lie algebras to these categories and show they form an equivalence of $\infty$-categories.

Global Koszul Duality: Differential Graded Cocommutative Coalgebras and Curved Lie Algebras

Abstract

We give a combinatorial model structure to the category of, not necessarily conilpotent, differential graded (dg) cocommutative coalgebras and an -category structure to the category of curved Lie algebras over an algebraically closed field of characteristic . Further, we extend the Harrison and Chevally-Eilenberg functors between dg cocommutative conilpotent coalgebras and dg Lie algebras to these categories and show they form an equivalence of -categories.
Paper Structure (11 sections, 19 theorems, 31 equations)

This paper contains 11 sections, 19 theorems, 31 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions.
  3. Outline of the paper
  4. Acknowledgements.
  5. Extended Chevalley-Eilenberg functors
  6. Commutative pseudocompact algebras
  7. Maurer-Cartan elements and representability
  8. Curved Lie algebras and the corresponding extended Chevalley-Eilenberg Functors
  9. The Model Structure on cocommutative coalgebras
  10. Model category structure on cocommutative coalgebras
  11. Koszul duality between cocommutative coalgebras and curved Lie algebras

Key Result

Proposition 2.2

Let $V$ be a pseudocompact graded vector space.

Theorems & Definitions (55)

  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Proposition 2.8
  • ...and 45 more