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We define a graded twisted-coassociative coproduct on the tensor algebra $TW$of any $\Z^n$-graded vector space $W$. If $W$ is the desuspension space $\da V$of a graded vector space $V$, the coderivations (resp. quadratic ``degree 1''codifferentials, arbitrary odd codifferentials) of this coalgebra are 1-to-1with 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 infinitystructures on $V$). We prove a minimal model theorem for Loday infinityalgebras, investigate Loday infinity morphisms, and observe that the$\op{Lod}_{\infty}$ category contains the $\op{L}_{\infty}$ category as asubcategory. Moreover, the graded Lie bracket of coderivations gives rise to agraded Lie ``stem'' bracket on the cochain spaces of graded Loday, Lodayinfinity, and $2n$-ary graded Loday algebras (the latter extend thecorresponding Lie algebras in the sense of Michor and Vinogradov). Thesealgebraic structures have square zero with respect to the stem bracket, so thatwe obtain natural cohomological theories that have good properties with respectto formal deformations. The stem bracket restricts to the gradedNijenhuis-Richardson and--up to isomorphism--to the Grabowski-Marmo brackets(the last bracket extends the Schouten-Nijenhuis bracket to the space of gradedantisymmetric first order polydifferential operators), and it encodes, beyondthe already mentioned cohomologies, those of graded Lie, graded Poisson, gradedJacobi, Lie infinity, as well as that of $2n$-ary graded Lie algebras.

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