Types of quantum field thories
The infinitesimal approximation to this smooth groupoid is a Lie algebroid. The dg-algebra of functions on this is called the BRST complex of the gauge theory. It contains in degree-0 the (duals to) the gauge fields and in degree-1 the cotangents to the gauge transformations. These degree-1 elements that appear here alongside the physical fields in degree 0 are called ghost fields in the physics literature.
If there are higher gauge transformations “gauge-of-gauge transformations” then the BRST complex has generators in higher degree, too, the cotangents to these higher gauge transformations. These are then called ghost-of-ghost fields.