Coalgebraic Approach to the Loday Infinity Category, Stem Differential for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> -ary Graded and Homotopy Algebras

Abstract

We define a graded twisted-coassociative coproduct on the tensor algebra the desuspension space of a graded vector space <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> . The coderivations (resp. quadratic “degree 1” codifferentials, arbitrary odd codifferentials) of this coalgebra are 1-to-1 with sequences of multilinear maps on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> (resp. graded Loday structures on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> , sequences that we call Loday infinity structures on <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>V</mml:mi> </mml:math> ). We prove a minimal model theorem for Loday infinity algebras and observe that the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>Lod</mml:mtext> <mml:mrow> <mml:mspace width="-0.166667em"/> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msub> </mml:math> category contains the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mtext>L</mml:mtext> <mml:mrow> <mml:mspace width="-0.166667em"/> <mml:mi>∞</mml:mi> </mml:mrow> </mml:msub> </mml:math> category as a subcategory. Moreover, the graded Lie bracket of coderivations gives rise to a graded Lie “stem” bracket on the cochain spaces of graded Loday, Loday infinity, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> -ary graded Loday algebras. This stem bracket restricts to the graded Nijenhuis-Richardson and Grabowski-Marmo brackets, and it encodes, beyond the already mentioned cohomologies, those of graded Lie, graded Poisson, graded Jacobi, Lie infinity, as well as that of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:math> -ary graded Lie algebras.