Categorification of $ \mathsf{VB} $-Lie algebroids and $ \mathsf{VB} $-Courant algebroids

In this paper, first we introduce the notion of a $ \mathsf{VB} $-Lie $ 2 $-algebroid, which can be viewed as the categorification of a $ \mathsf{VB} $-Lie algebroid. The tangent prolongation of a Lie $ 2 $-algebroid is a $ \mathsf{VB} $-Lie $ 2 $-algebroid naturally. We show that after choosing a splitting, there is a one-to-one correspondence between $ \mathsf{VB} $-Lie $ 2 $-algebroids and flat superconnections of a Lie 2-algebroid on a 3-term complex of vector bundles. Then we introduce the notion of a $ \mathsf{VB} $-$ \mathsf{CLWX} $ 2-algebroid, which can be viewed as the categorification of a $ \mathsf{VB} $-Courant algebroid. We show that there is a one-to-one correspondence between split Lie 3-algebroids and split $ \mathsf{VB} $-$ \mathsf{CLWX} $ 2-algebroids. Finally, we introduce the notion of an $ E $-$ \mathsf{CLWX} $ 2-algebroid and show that associated to a $ \mathsf{VB} $-$ \mathsf{CLWX} $ 2-algebroid, there is an $ E $-$ \mathsf{CLWX} $ 2-algebroid structure on the graded fat bundle naturally. By this result, we give a construction of a new Lie 3-algebra from a given Lie 3-algebra, which provides interesting examples of Lie 3-algebras including the higher analogue of the string Lie 2-algebra.