Categorification of VB-Lie algebroids and VB-Courant algebroids

Abstract

In this paper, first we introduce the notion of a -Lie $2$-algebroid, which can be viewed as the categorification of a -Lie algebroid. The tangent prolongation of a Lie $2$-algebroid is a -Lie $2$-algebroid naturally. We show that after choosing a splitting, there is a one-to-one correspondence between -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 - 2-algebroid, which can be viewed as the categorification of a -Courant algebroid. We show that there is a one-to-one correspondence between split Lie 3-algebroids and split - 2-algebroids. The notion of a -Lie $2$