Abstract
In this paper, we develop the A_infty -analog of the Maurer-Cartan simplicial set associated to an L_infty -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 -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 -algebras to simplicial sets, which sends a complete curved A_infty -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.
Highlights
One important example of an application of the Maurer-Cartan simplicial set of an L∞-algebra, is that the Maurer-Cartan simplicial set can be used in deformation theory to encode the set of deformations of an object
The theory developed in this paper, tries to solve this gap in the case that the deformation problem is controlled by an A∞-algebra
When A and B are filtered, we will always assume that ∞-morphisms respect the filtrations in this way and call this an ∞-imorphism
Summary
2, we introduce the necessary preliminaries on coassociative coalgebras, like filtrations and the shuffle product, which we need to define A∞-algebras. 3, we give the definition of A∞-algebras and define their morphisms. 4, we describe the twist of an A∞-algebra using the shuffle product and define the Maurer-Cartan equation. 5, we prove a few technical lemmas which are important for Sect. 6. In that section we define the Maurer-Cartan simplicial set associated to an A∞-algebra and show that it is a Kan complex. 7, we apply the theory developed in this paper to the deformation theory of ∞-morphism of algebras over non-symmetric operads. We finish this paper with a comparison with other approaches, some possible directions for future work, and some open questions
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