On the Maurer-Cartan simplicial set of a complete curved A∞-algebra
| Autor: | Niek de Kleijn, Felix Wierstra |
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| Přispěvatelé: | Algebra, Geometry & Mathematical Physics (KDV, FNWI) |
| Jazyk: | angličtina |
| Rok vydání: | 2021 |
| Předmět: |
Computer Science::Machine Learning
Algebra and Number Theory Functor Field (mathematics) Algebraic topology Computer Science::Digital Libraries Mathematics::Algebraic Topology Algebra Statistics::Machine Learning Number theory Morphism Mathematics::K-Theory and Homology Product (mathematics) Mathematics::Quantum Algebra Mathematics::Category Theory Computer Science::Mathematical Software Simplicial set Geometry and Topology Twist Mathematics |
| Zdroj: | Journal of Homotopy and Related Structures, 16(4), 605-633. Springer Science + Business Media Journal of Homotopy and Related Structures volume Journal of Homotopy and Related Structures, 16(4) |
| ISSN: | 2193-8407 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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