Homotopy morphisms between convolution homotopy Lie algebras

Robert-Nicoud, Daniel; Wierstra, Felix

doi:10.4171/jncg/351

Cited by 6 publications

(10 citation statements)

References 19 publications

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“…This paper concludes a series of articles by the two authors dealing with the investigation of convolution algebras which started with [Wie16], and [RN18a], and then continued jointly with [RNW17]. of the underlying chain complexes.…”

Section: Introductionmentioning

confidence: 85%

“…Having extended the bifunctor hom α (−, −) to the two bifunctors hom α r (−, −) and hom α ℓ (−, −) accepting ∞ α -morphisms in the right and left slot respectively, it is natural to ask if those two functors admit a common extension to a bifunctor accepting ∞ α -morphisms in both slots simultaneously. Unfortunately, this is not possible, as was proven by the authors in [RNW17,Sect. 6].…”

Section: The Two Bifunctors Commute Up To Homotopymentioning

confidence: 98%

“…There is one important but straightforward fact that was not proven in [RNW17], which relates the morphisms hom α r (1, −) and hom α ℓ (−, 1), and compositions of morphisms. Recall that the action of an ∞-morphisms Θ : g → h of sL ∞ -algebras on Maurer-Cartan elements is given by…”

Section: Infinity-morphisms and Two Bifunctorsmentioning

confidence: 99%

“…We will use essentially the same notation and conventions as in [RNW17]. By transitivity, we will follow the notation of the book [LV12] as closely as possible when talking about operads.…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…These algebras have already found various applications. They helped to construct a "universal Maurer-Cartan element" in [RN17], they are used to construct complete rational invariants of maps between topological spaces in [Wie16], and they were applied to the construction of rational models for mapping spaces in [RNW17].…”

Section: Introductionmentioning

confidence: 99%

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Convolution algebras and the deformation theory of infinity-morphisms

Robert-Nicoud

Wierstra

2019

Homology, Homotopy and Applications

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Given a coalgebra C over a cooperad and an algebra A over an operad, it is often possible to define a natural homotopy Lie algebra structure on hom(C, A), the space of linear maps between them, called the convolution algebra of C and A. In the present article, we use convolution algebras to define the deformation complex for ∞-morphisms of algebras over operads and coalgebras over cooperads. We also complete the study of the compatibility between convolution algebras and ∞-morphisms of algebras and coalgebras. We prove that the convolution algebra bifunctor can be extended to a bifunctor that accepts ∞morphisms in both slots and which is well defined up to homotopy, and we generalize and take a new point of view on some other already known results. This paper concludes a series of works by the two authors dealing with the investigation of convolution algebras. Contents 1 Introduction 352 2 Infinity-morphisms and convolution algebras 354 3 Compatibility between convolution algebras and infinity-morphisms 357 4 Proof of Theorems 2.4, 3.1, and 3.6 362 A A counterexample 370

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Section: Introductionmentioning

confidence: 85%

Section: The Two Bifunctors Commute Up To Homotopymentioning

confidence: 98%

Section: Infinity-morphisms and Two Bifunctorsmentioning

confidence: 99%

“…We will use essentially the same notation and conventions as in [RNW17]. By transitivity, we will follow the notation of the book [LV12] as closely as possible when talking about operads.…”

Section: Acknowledgmentsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Convolution algebras and the deformation theory of infinity-morphisms

Robert-Nicoud

Wierstra

2019

Homology, Homotopy and Applications

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On the Maurer-Cartan simplicial set of a complete curved $$A_\infty $$-algebra

Kleijn

Wierstra

2021

J. Homotopy Relat. Struct.

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In this paper, we develop the $$A_\infty $$ A ∞ -analog of the Maurer-Cartan simplicial set associated to an $$L_\infty $$ L ∞ -algebra and show how we can use this to study the deformation theory of $$\infty $$ ∞ -morphisms of algebras over non-symmetric operads. More precisely, we first recall and prove some of the main properties of $$A_\infty $$ A ∞ -algebras like the Maurer-Cartan equation and twist. One of our main innovations here is the emphasis on the importance of the shuffle product. Then, we define a functor from the category of complete (curved) $$A_\infty $$ A ∞ -algebras to simplicial sets, which sends a complete curved $$A_\infty $$ A ∞ -algebra to the associated simplicial set of Maurer-Cartan elements. This functor has the property that it gives a Kan complex. In all of this, we do not require any assumptions on the field we are working over. We also show that this functor can be used to study deformation problems over a field of characteristic greater than or equal to 0. As a specific example of such a deformation problem, we study the deformation theory of $$\infty $$ ∞ -morphisms of algebras over non-symmetric operads.

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The integration theory of curved absolute L∞-algebras

Roca i Lucio

2024

Advances in Mathematics

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Homotopy morphisms between convolution homotopy Lie algebras

Cited by 6 publications

References 19 publications

Convolution algebras and the deformation theory of infinity-morphisms

Convolution algebras and the deformation theory of infinity-morphisms

On the Maurer-Cartan simplicial set of a complete curved $$A_\infty $$-algebra

The integration theory of curved absolute L∞-algebras

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