Convolution algebras and the deformation theory of infinity-morphisms

Robert-Nicoud, Daniel; Wierstra, Felix

doi:10.4310/hha.2019.v21.n1.a17

Cited by 3 publications

(5 citation statements)

References 16 publications

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“…element in the pre-Lie algebra associated to the convolution operad. The following theorem is the non-symmetric analog of Lemma 4.1 and Theorem 7.1 of [32], see also Section 4 of [26].…”

Section: Corollary 75mentioning

confidence: 94%

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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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“…element in the pre-Lie algebra associated to the convolution operad. The following theorem is the non-symmetric analog of Lemma 4.1 and Theorem 7.1 of [32], see also Section 4 of [26].…”

Section: Corollary 75mentioning

confidence: 94%

“…The main advantage of our theory is, that everything now also works over a field of arbitrary characteristic and not just over a field of characteristic 0. Most of this section is the nonsymmetric version of the results of[26] and[27]. Since all the proofs are completely analogous to the proofs in those papers, we will omit most of them.…”

mentioning

confidence: 97%

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

Kleijn

Wierstra

2021

J. Homotopy Relat. Struct.

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“…We also expect future applications of our work to the computation of the homology of fibered spaces, using the construction of the convolution A ∞ -algebra associated to an A ∞ -coalgebra and an A ∞ -algebra in Proposition 5.4. This last construction can also be related to the deformation theory of ∞-morphisms developed in [RNW19b,RNW19a], see Section 5.2.3. Moreover, our geometric methods shed a new light on a result of M. Markl and S. Shnider [MS06], pointing towards possible links with discrete and continuous Morse theory (Remark 5.3).…”

Section: Introductionmentioning

confidence: 94%

“…We will use this key property in order to pursue the work of Brown [Bro59] and [Pro86] on the homology of fibered spaces in a forthcoming paper. The main result of [RNW19a] says that if a twisting morphism α ∈ Tw(BAs, ΩAs ¡ ) is Koszul, then the possible compositions of the two bifunctors (5.1) and (5.2) are homotopic and that they extend to a bifunctor on the level of the homotopy categories [RNW19a,Th. 3.6 and Cor.…”

Section: Proofmentioning

confidence: 99%

“…This should be seen as a statement analogous to Point (3) of Proposition 4.26. It would be interesting to know how the results of [RNW19b,RNW19a] can be interpreted from the viewpoint of diagonals, and if they admit an interpretation on the level of polytopes. Thurston also study diagonals on the dg operad A ∞ and on the dg operadic bimodule M ∞ .…”

Section: Proofmentioning

confidence: 99%

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The diagonal of the multiplihedra and the tensor product of A ∞ -morphisms

Laplante-Anfossi¹,

Mazuir²

2023

Journal De L’École Polytechnique — Mathématiques

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We define a cellular approximation for the diagonal of the Forcey-Loday realizations of the multiplihedra, and endow them with a compatible topological cellular operadic bimodule structure over the Loday realizations of the associahedra. This provides us with a model for topological and algebraic A∞-morphisms, as well as a universal and explicit formula for their tensor product. We study the monoidal properties of this newly defined tensor product and conclude by outlining several applications, notably in algebraic and symplectic topology. Résumé (La diagonale des multiplièdres et le produit tensoriel de morphismes A-infini)On définit une approximation cellulaire de la diagonale des réalisations de Forcey-Loday des multiplièdres, et on les munit d'une structure de bimodule opéradique topologique et cellulaire compatible sur les réalisations de Loday des associaèdres. On obtient ainsi un modèle algébrique et topologique pour les morphismes A-infini, de même qu'une formule universelle explicite pour leur produit tensoriel. On étudie la monoïdalité de ce nouveau produit tensoriel et on conclut en esquissant plusieurs applications en topologie algébrique et en topologie symplectique.

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Algebraic Hopf invariants and rational models for mapping spaces

Wierstra

2019

J. Homotopy Relat. Struct.

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The main goal of this paper is to define an invariant mc∞(f ) of homotopy classes of maps f : X → Y Q , from a finite CW-complex X to a rational space Y Q . We prove that this invariant is complete, i.e. mc∞(f ) = mc∞(g) if an only if f and g are homotopic.To construct this invariant we also construct a homotopy Lie algebra structure on certain convolution algebras. More precisely, given an operadic twisting morphism from a cooperad C to an operad P, a C-coalgebra C and a P-algebra A, then there exists a natural homotopy Lie algebra structure on Hom K (C, A), the set of linear maps from C to A. We prove some of the basic properties of this convolution homotopy Lie algebra and use it to construct the algebraic Hopf invariants. This convolution homotopy Lie algebra also has the property that it can be used to models mapping spaces. More precisely, suppose that C is a C∞-coalgebra model for a simply-connected finite CWcomplex X and A an L∞-algebra model for a simply-connected rational space Y Q of finite Q-type, then Hom K (C, A), the space of linear maps from C to A, can be equipped with an L∞-structure such that it becomes a rational model for the based mapping space M ap * (X, Y Q ).

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Convolution algebras and the deformation theory of infinity-morphisms

Cited by 3 publications

References 16 publications

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

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

The diagonal of the multiplihedra and the tensor product of A ∞ -morphisms

Algebraic Hopf invariants and rational models for mapping spaces

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