  In this paper, we introduce the notions of hom-Lie 2-algebras, which is the
categorification of hom-Lie algebras, $HL_\infty$-algebras, which is the
hom-analogue of $L_\infty$-algebras, and crossed modules of hom-Lie algebras.
We prove that the category of hom-Lie 2-algebras and the category of 2-term
$HL_\infty$-algebras are equivalent. We give a detailed study on skeletal
hom-Lie 2-algebras. In particular, we construct the hom-analogues of the string
Lie 2-algebras associated to any semisimple involutive hom-Lie algebras. We
also proved that there is a one-to-one correspondence between strict hom-Lie
2-algebras and crossed modules of hom-Lie algebras. We give the construction of
strict hom-Lie 2-algebras from hom-left-symmetric algebras and symplectic
hom-Lie algebras.
