Categorification of VB-Lie algebroids and VB-Courant algebroids

In this paper, first we introduce the notion of a $\VB$-Lie $2$-algebroid, which can be viewed as the categorification of a $\VB$-Lie algebroid. The tangent prolongation of a Lie $2$-algebroid is a $\VB$-Lie $2$-algebroid naturally. We show that after choosing a splitting, there is a one-to-one correspondence between $\VB$-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 $\VB$-$\LWX$ 2-algebroid, which can be viewed as the categorification of a $\VB$-Courant algebroid. We show that there is a one-to-one correspondence between split Lie 3-algebroids and split $\VB$-$\LWX$ 2-algebroids. The notion of a $\VB$-Lie $2$-bialgebroid is introduced and the double of a $\VB$-Lie $2$-bialgebroid is a $\VB$-$\LWX$ 2-algebroid. Finally, we introduce the notion of an $E$-$\LWX$ 2-algebroid and show that associated to a $\VB$-$\LWX$ 2-algebroid, there is an $E$-$\LWX$ 2-algebroid structure on the graded fat bundle naturally. By this result, we give a construction of a 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.