Higher structures for Lie H-pseudoalgebras

Let H be a cocommutative Hopf algebra. The notion of Lie H-pseudoalgebra is a multivariable generalization of Lie conformal algebras. In this paper, we study some higher structures related to Lie H-pseudoalgebras where we increase the flexibility of the Jacobi identity. Namely, we first introduce L_ H-pseudoalgebras (also called strongly homotopy Lie H-pseudoalgebras) as the homotopy analogue of Lie H-pseudoalgebras. We give several equivalent descriptions of such homotopy algebras and show that some particular classes of these homotopy algebras are closely related to the cohomology of Lie H-pseudoalgebras and crossed modules of Lie H-pseudoalgebras. Next, we introduce another higher structure, called Lie-2 H-pseudoalgebras which are the categorification of Lie H-pseudoalgebras. Finally, we show that the category of Lie-2 H-pseudoalgebras is equivalent to the category of certain L_ H-pseudoalgebras.