Algebraic Hopf invariants and rational models for mapping spaces

Wierstra, Felix

doi:10.1007/s40062-018-00230-z

Cited by 8 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%

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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%

“…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%

Convolution algebras and the deformation theory of infinity-morphisms

Robert-Nicoud

Wierstra

2019

Homology, Homotopy and Applications

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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%

“…Using the fact that when we have a C-coalgebra C and a P-algebra A, then Hom(C, A) is a Hom(C, P)-algebra (see Proposition 7.1 of [32]), we have the following corollary.…”

Section: Corollary 75mentioning

confidence: 99%

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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“…Remark 5.0.1. In recent work by Robert-Nicoud and Wierstra [61,70], it was shown that-given a twisting morphism 𝛼: 𝒞 → 𝒫 as above-one can construct an L ∞ -algebra structure on Conv(𝐶, 𝐴). Maurer-Cartan elements of this L ∞ -algebra Conv 𝛼 (𝐶, 𝐴) correspond to twisting morphisms Tw 𝛼 (𝐶, 𝐴) as defined above.…”

Section: Resolutions Of Algebrasmentioning

confidence: 99%

Symmetric Homotopy Theory for Operads and Weak Lie 3-Algebras

Dehling¹

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Sciences for the invitations and the excellent working conditions. The author would also like to thank Chris Rogers and Dmitry Roytenberg for their insightful remarks and Yunhe Sheng for his inspiration. The author is most grateful to Chenchang Zhu for many helpful discussions and suggestions over the years, and to Bruno Vallette for his collaboration, valuable insights, and hospitality on numerous occasions. CHAPTER 1 PreliminariesThe aim of this chapter is to introduce notation and conventions used throughout this thesis.We denote by 𝐤 an arbitrary commutative unital ring and by 𝐤-Mod its category of modules. In practice, for any computations we will work over the integers 𝐤 = ℤ. Since ℤ is the initial object in the category of unital commutative rings, this ensures that our results hold over any such ring 𝐤. Chain ComplexesThe index 𝑖 is referred to as (homological) degree, and we use the notation |𝑣| = 𝑖 for elements 𝑣 ∈ 𝑉 𝑖 . A morphism of chain complexes in 𝐤-modules 𝑓: (𝑉, d 𝑉 ) → (𝑊, d 𝑊 ) is a collection of 𝐤-linear maps {𝑓 𝑖 : 𝑉 𝑖 → 𝑊 𝑖 } 𝑖∈ℤ , such that d 𝑊 𝑖 ∘ 𝑓 𝑖 = 𝑓 𝑖−1 ∘ d 𝑉 𝑖 for all 𝑖 ∈ ℤ. The category of chain complexes in 𝐤-Mod is denoted by 𝐤-Ch. 1.1. Abelian structure. It is a standard result that 𝐤-Ch is an abelian category with biproduct the degreewise direct sum of 𝐤-modules (𝑉 ⊕ 𝑊) 𝑖 ≔ 𝑉 𝑖 ⊕ 𝑊 𝑖 equipped with the componentwise differential d 𝑉 ⊕𝑊 (𝑣 + 𝑤) = d 𝑉 (𝑣) + d 𝑊 (𝑤). Kernels and cokernels are computed degreewise, i.e., ker(𝑓) 𝑖 = ker(𝑓 𝑖 ) and coker(𝑓) 𝑖 = coker(𝑓 𝑖 ). Symmetric monoidal structure.The category 𝐤-Ch can be equipped with a monoidal productSlightly abusing notation, the chain complex given by 𝐤 in degree 0 and 0 in all other degrees is denoted again by 𝐤. It acts as the unit object with respect to the monoidal product. The monoidal product satisfies a certain symmetry: we denote by 𝜏 the natural isomorphism with components given on homogeneous elementary tensors by (2) 𝜏 𝑉 ,𝑊 : 𝑉 ⊗ 𝑊 ⟶ 𝑊 ⊗ 𝑉 , 𝑣 ⊗ 𝑤 ⟼ (−1) |𝑣||𝑤| ⋅ 𝑤 ⊗ 𝑣 .Clearly 𝜏 𝑉 ,𝑊 ∘ 𝜏 𝑊 ,𝑉 = id, and together with the above monoidal structure this turns (𝐤-Ch, ⊗, 𝐤) into a symmetric monoidal category.Remark 1.2.1. The sign in the differential of the tensor product in Equation ( 1) is necessary to ensure that d 𝑉 ⊗𝑊 squares to zero, and, as a consequence, the sign in the symmetry isomorphism in Equation ( 2) is required such that the components of 𝜏 are 5 𝜒:𝐵→𝑚

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Algebraic Hopf invariants and rational models for mapping spaces

Cited by 8 publications

References 21 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

Symmetric Homotopy Theory for Operads and Weak Lie 3-Algebras

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