Hom-Leibniz 2-algebras

Abstract

We define a Hom-Leibniz 2-algebra, which can be considered as the deformation and categorification of Leibniz algebras. We give the notion of 2-term strongly homotopy (sh) Hom-Leibniz algebras. We prove that the 2-term sh Hom-Leibniz algebras are equivalent to the Hom-Leibniz 2-algebras. We show that there exists a one-to-one correspondence between crossed modules of Hom-Leibniz algebras and 2-term Hom-differential graded Leibniz algebras. We also prove that the skeletal 2-term sh Hom-Leibniz algebras can be classified by using the third cohomology group of Hom-Leibniz algebras. We construct a Hom-Leibniz 2-algebra from an Omni-Hom-Lie 2-algebra. Moreover, we also give the construction of Hom-Leibniz 2-algebras from Hom-associative Rota–Baxter algebras.