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

摘要: We define a graded twisted-coassociative coproduct on the tensor algebra TW of anyZ n -graded vector space W . If is desuspension # V , coderivations (resp. quadratic “degree 1” codifferentials, arbitrary odd codifferentials) this coalgebra are 1to-1 with sequences p s, s ‚ 1, s-linear maps Z Loday structures that we call infinity ). prove minimal model theorem for algebras, investigate morphisms, and observe Lod¥ category contains L¥ as subcategory. Moreover, Lie bracket gives rise to “stem” cochain spaces Loday, infinity, 2n-ary algebras (the latter extend corresponding in sense Michor Vinogradov). These algebraic have square zero respect stem bracket, so obtain natural cohomological theories good properties formal deformations. The restricts Nijenhuis-Richardson and— up isomorphism—to Grabowski-Marmo brackets last extends SchoutenNijenhuis antisymmetric first order polydifferential operators), it encodes, beyond already mentioned cohomologies, those Lie, Poisson, Jacobi, well algebras.