Cocommutative coalgebras: homotopy theory and Koszul duality

Chuang, Joseph; Lazarev, Andrey; Mannan, W. H.

doi:10.48550/arxiv.1403.0774

Cited by 4 publications

(24 citation statements)

References 12 publications

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“…cit., and thus the homotopy theory developed therein is not completely general. Furthermore, Positselski worked with curved objects suggesting that when discussing a Koszul duality in more general cases one side of the Quillen equivalence should be a category consisting of curved objects; a hypothesis that is strengthened by results of [CLM14] and this paper.…”

Section: Introductionmentioning

confidence: 63%

“…In the case of a strict morphism it can be readily seen that the morphism is simply a graded Lie algebra morphism that respects the differentials and the image of the curvature of the domain is the curvature of the codomain. These morphisms are exactly those of [CLM14]. The α part of a curved morphism can, therefore, be seen to act as an obstruction to the differentials commuting with the graded Lie algebra morphism and to the graded Lie algebra morphism preserving the curvature.…”

Section: The Category Of Curved Lie Algebrasmentioning

confidence: 99%

“…This becomes a unital cdga with the concatenation product and the differential made of three parts coming from the duals of the curvature, the differential and the bracket of g made into derivations and extended via the Leibniz rule, cf. [CLM14].…”

Section: Analogues Of the Chevalley-eilenberg And Harrison Complexesmentioning

confidence: 99%

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Unbased rational homotopy theory: a Lie algebra approach

Maunder¹

2015

Preprint

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In this paper an algebraic model for unbased rational homotopy theory from the perspective of curved Lie algebras is constructed. As part of this construction a model structure for the category of pseudocompact curved Lie algebras with curved morphisms will be introduced; one which is Quillen equivalent to a certain model structure of unital commutative differential graded algebras, thus extending the known Quillen equivalence of augmented algebras and differential graded Lie algebras.

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Section: Introductionmentioning

confidence: 63%

Section: The Category Of Curved Lie Algebrasmentioning

confidence: 99%

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Unbased rational homotopy theory: a Lie algebra approach

Maunder¹

2015

Preprint

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“…cit. When the ground field is algebraically closed, the construction was completely generalised to all coalgebras by Chuang, Lazarev, and Mannan [5]. The authors therein chose to work in the dual setting of pseudo-compact unital commutative differential graded algebras.…”

Section: Introductionmentioning

confidence: 99%

“…The Koszul duality is also extended in [5]. The Koszul dual therein is the category of formal coproducts of curved Lie algebras.…”

Section: Introductionmentioning

confidence: 99%

Koszul duality and homotopy theory of curved Lie algebras

Maunder

2017

Homology, Homotopy and Applications

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Abstract. This paper introduces the category of marked curved Lie algebras with curved morphisms, equipping it with a model structure. This model structure is-when working over an algebraically closed field of characteristic zero-Quillen equivalent to a model category of pseudo-compact unital commutative differential graded algebras; extending known results regarding the Koszul duality of unital commutative differential graded algebras and differential graded Lie algebras. As an application of the theory developed within this paper, algebraic deformation theory is extended to functors over pseudocompact, not necessarily local, commutative differential graded algebras. Further, these deformation functors are shown to be representable.

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Double field theory algebroid and curved L ∞-algebras

Grewcoe

Jonke

2021

Journal of Mathematical Physics

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A double field theory algebroid (DFT algebroid) is a special case of the metric (or Vaisman) algebroid, shown to be relevant in understanding the symmetries of double field theory. In particular, a DFT algebroid is a structure defined on a vector bundle over doubled spacetime equipped with the C-bracket of double field theory. In this paper, we give the definition of a DFT algebroid as a curved L∞-algebra and show how implementation of the strong constraint of double field theory can be formulated as an L∞-algebra morphism. Our results provide a useful step toward coordinate invariant descriptions of double field theory and the construction of the corresponding sigma-model.

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Cocommutative coalgebras: homotopy theory and Koszul duality

Cited by 4 publications

References 12 publications

Unbased rational homotopy theory: a Lie algebra approach

Unbased rational homotopy theory: a Lie algebra approach

Koszul duality and homotopy theory of curved Lie algebras

Double field theory algebroid and curved L ∞-algebras

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