Properadic Homotopical Calculus

Hoffbeck, Eric; Leray, Johan; Vallette, Bruno

doi:10.1093/imrn/rnaa091

Cited by 5 publications

(6 citation statements)

References 15 publications

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“…Let P : A → A be a Hodge propagator and H ⊂ A the associated harmonic subspace. We identify the cyclic cochain complexes (H, d = 0, Inserting ( 27) and ( 28) in (24) leads to an explicit description of the pushforward Maurer-Cartan element (f P * m can A ) k, ,g in terms of trivalent ribbon graphs:…”

Section: Chain-level Equivariant String Topology For Simply Connected...mentioning

confidence: 99%

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Chain-level equivariant string topology for simply connected manifolds

Cieliebak¹,

Hájek²,

Волков³

2022

Preprint

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We prove that on 2-connected closed oriented manifolds, the analytic and algebraic constructions of chain-level equivariant string topology coincide. The corresponding structure is invariant under orientation preserving homotopy equivalences and induces on homology the involutive Lie bialgebra structure of Chas and Sullivan. Contents 1. Introduction 2. Poincaré DGAs 2.1. Cochain complexes with pairing 2.2. Differential graded algebras with pairing 2.3. Poincaré DGAs and differential Poincaré duality models 3. IBL ∞ algebras 3.1. Basic definitions and properties 3.2. The dIBL structure associated to a cyclic cochain complex 3.3. Weak functoriality of the dIBL construction 3.4. The twisted dIBL structure associated to a cyclic DGA 4. The algebraic construction 4.1. Algebraic vanishing results 4.2. Weak functoriality of the twisted dIBL construction 4.3. Twisted dIBL algebras associated to Poincaré DGAs 5. The analytic construction 5.1. Analytic propagators 5.2. The analytic Maurer-Cartan element 5.3. Analytic vanishing results 6. Comparison of the algebraic and analytic constructions 6.1. A ∞ algebras and homotopy transfer 6.2. Cyclic A ∞ algebras and IBL ∞ Maurer Cartan elements 6.3. Comparison of the algebraic and analytic Maurer Cartan elements 6.4. Formality 7. Relation to equivariant string topology 7.1. Reduced version 7.2. Chain model of Chas-Sullivan IBL algebra

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Section: Chain-level Equivariant String Topology For Simply Connected...mentioning

confidence: 99%

“…We recall this notion in Section 3 below. See also [24] for further discussion of IBL ∞ algebras from a properadic perspective.…”

Section: Introductionmentioning

confidence: 99%

Chain-level equivariant string topology for simply connected manifolds

Cieliebak¹,

Hájek²,

Волков³

2022

Preprint

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“…T(X)((2)) and T(X)((1)) ⊗ T(X)(( 1)). This can be fixed by also remembering the "horizontal composition" of T(X), see work of Hoffbeck-Leray-Vallette [HLV,§3.1]. (See also Remark 2.18.…”

Section: Lax Algebrasmentioning

confidence: 99%

Modular operads as modules over the Brauer properad

Stoll¹

2022

Preprint

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We show that modular operads are equivalent to modules over a certain simple properad which we call the Brauer properad. Furthermore we show that, in this setting, the Feynman transform corresponds to the cobar construction for modules of this kind. To make this precise, we extend the machinery of the bar and cobar constructions relative to a twisting morphism to modules over a general properad. This generalizes the classical case of algebras over an operad and might be of independent interest. As an application, we sketch a Koszul duality theory for modular operads.

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“…3. Homotopy transfer for IBL ∞ algebras IBL ∞ algebras (short for Involutive Lie Bialgebras up to coherent homotopies) were introduced in the paper [21], with applications to string topology, symplectic field theory and Lagrangian Floer theory, and have been further investigated in several other papers since then, for instance [18,57,25,39,58,41,59,22,40].…”

Section: Proof Denoting By P : S(b[[t]]) → B[[t]mentioning

confidence: 99%

“…IBL ∞ algebras (short for Involutive Lie Bialgebras up to coherent homotopies) were introduced in the paper [21], with applications to string topology, symplectic field theory and Lagrangian Floer theory, and have been further investigated in several other papers since then, for instance [18,57,25,39,58,41,59,22,40] We shall work with a definition of IBL ∞ algebra slightly different from, and in a certain sense dual to, the one usually appearing in the literature. More precisely, whereas IBL ∞ algebras are usually regarded as commutative BV ∞ algebras whose underlying algebras are free (see for instance [20,18,57]), we shall regard them dually as cocommutative BV ∞ coalgebras whose underlying coalgebras are cofree.…”

Section: Introductionmentioning

confidence: 99%

Cumulants, Koszul brackets and homological perturbation theory for commutative $BV_\infty$ and $IBL_\infty$ algebras

Bandiera¹

2020

Preprint

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We explore the relationship between the classical constructions of cumulants and Koszul brackets, showing that the former are an expontial version of the latter. Moreover, under some additional technical assumptions, we prove that both constructions are compatible with standard homological perturbation theory in an appropriate sense. As an application of these results, we provide new proofs for the homotopy transfer Theorem for L∞ and IBL∞ algebras based on the symmetrized tensor trick and the standard perturbation Lemma, as in the usual approach for A∞ algebras. Moreover, we prove a homotopy transfer Theorem for commutative BV∞ algebras in the sense of Kravchenko which appears to be new. Along the way, we introduce a new definition of morphism between commutative BV∞ algebras. Contents 37 3.1. IBL ∞ [1] algebras 37 3.2. Homotopy transfer for IBL ∞ [1] algebras 40 References 42

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Properadic Homotopical Calculus

Cited by 5 publications

References 15 publications

Chain-level equivariant string topology for simply connected manifolds

Chain-level equivariant string topology for simply connected manifolds

Modular operads as modules over the Brauer properad

Cumulants, Koszul brackets and homological perturbation theory for commutative $BV_\infty$ and $IBL_\infty$ algebras

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