  In this work we solve the problem of providing a Morita invariant definition
of Lie and Courant algebroids over Lie groupoids. By relying on supergeometry,
we view these structures as instances of vector fields on graded groupoids
which are homological up to homotopy. We describe such vector fields in general
from two complementary viewpoints: firstly, as Maurer-Cartan elements in a
differential graded Lie algebra of multivector fields and, secondly, we also
view them from a categorical approach, in terms of functors and natural
transformations. Thereby, we obtain a unifying conceptual framework for
studying LA-groupoids, $L_2$-algebroids (including semistrict Lie 2-algebras
and 2-term representations up to homotopy), infinitesimal gerbe
prequantizations, higher gauge theory (specifically, 2-connections on
2-bundles), quasi-Poisson groupoids and (twisted) multiplicative Courant
algebroids.
