A universal characterization of the curved homotopy Lie and associative operads

Abstract

We study the category of nonsymmetric dg operads valued in strict graded-mixed complexes, equipped with a distinguished arity zero weight one element which generates the weight grading, and whose differential has weight one. We show that the initial object is the curved A-infinity operad, that the forgetful functor to the category of operads under it admits a right adjoint, and that the unit of the adjunction encodes the operation of twisting a curved A-infinity algebra by a Maurer-Cartan element. The corresponding notions for symmetric operads characterize the curved L-infinity operad and the corresponding twisting procedure.