Coalgebraic Approach to the Loday Infinity Category, Stem Differential for $2n$-ary Graded and Homotopy Algebras

We define a graded twisted-coassociative coproduct on the tensor algebra $TW$ of any ${\mathbb{Z}}^n$-graded vector space $W$. If $W$ is the desuspension space $\text{da}V$ of a graded vector space $V$, the coderivations (resp. quadratic degree 1'' codifferentials, arbitrary odd codifferentials) of this coalgebra are 1-to-1 with sequences $\zp_s$, $s\ge 1$, of $s$-linear maps on $V$ (resp. $\Z^n$-graded Loday structures on $V$, sequences that we call Loday infinity structures on $V$). We prove a minimal model theorem for Loday infinity algebras, investigate Loday infinity morphisms, and observe that the $\op{Lod}_{\infty}$ category contains the $\op{L}_{\infty}$ 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 $2n$-ary graded Loday algebras (the latter extend the corresponding Lie algebras in the sense of Michor and Vinogradov). These algebraic structures have square zero with respect to the stem bracket, so that we obtain natural cohomological theories that have good properties with respect to formal deformations. The stem bracket restricts to the graded Nijenhuis-Richardson and–up to isomorphism–to the Grabowski-Marmo brackets (the last bracket extends the Schouten-Nijenhuis bracket to the space of graded antisymmetric first order polydifferential operators), and it encodes, beyond the already mentioned cohomologies, those of graded Lie, graded Poisson, graded Jacobi, Lie infinity, as well as that of $2n$-ary graded Lie algebras.