TY - JOUR

T1 - On the Maurer-Cartan simplicial set of a complete curved A∞ -algebra

AU - de Kleijn, Niek

AU - Wierstra, Felix

PY - 2021

Y1 - 2021

N2 - In this paper, we develop the A∞-analog of the Maurer-Cartan simplicial set associated to an L∞-algebra and show how we can use this to study the deformation theory of ∞-morphisms of algebras over non-symmetric operads. More precisely, we first recall and prove some of the main properties of 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∞-algebras to simplicial sets, which sends a complete curved 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 ∞-morphisms of algebras over non-symmetric operads.

AB - In this paper, we develop the A∞-analog of the Maurer-Cartan simplicial set associated to an L∞-algebra and show how we can use this to study the deformation theory of ∞-morphisms of algebras over non-symmetric operads. More precisely, we first recall and prove some of the main properties of 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∞-algebras to simplicial sets, which sends a complete curved 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 ∞-morphisms of algebras over non-symmetric operads.

UR - http://www.scopus.com/inward/record.url?scp=85115696405&partnerID=8YFLogxK

U2 - 10.1007/s40062-021-00290-8

DO - 10.1007/s40062-021-00290-8

M3 - Article

AN - SCOPUS:85115696405

VL - 16

SP - 605

EP - 633

JO - Journal of Homotopy and Related Structures

JF - Journal of Homotopy and Related Structures

SN - 2193-8407

IS - 4

ER -