The notion of shifted tangent bundle makes sense also when itself was already a supermanifold. Superdifferential forms on a supermanifold are similarly the algebra of functions on the shifted tangent bundle .
Another way to think of superdifferential forms is using the perspective of Lie theory:
The algebra of superdifferential forms on is the exterior algebra generated over from
a generator in odd degree (the canonical odd coordinate);
a generator in odd degree (the differential of the canonical even coordinate);
a generator in even degree (the differential of the canonical odd coordinate).
Notice in particular that while the wedge product is non-vanishing, since is in even degree. In fact all higher wedge powers of with itself exist.
Being a -graded locally free algebra itself, one can regard itself (even for a usual manifold!) as the “algebra of functions” (more precisely inner hom, i.e. mapping space into the line) on another supermanifold. That supermanifold is called , the shifted tangent bundle of . By definition we have . From this point of view, the existence of the differential on the graded algebra translates into the existence of a special odd vector field on . This is a homological vector field in that it is odd and the super Lie bracket of it with itself vanishes: .
In the context of L-infinity algebroids, where one may regard as the Chevalley-Eilenberg algebra of an -algebroid it is useful to notice that is the corresponding Weil algebra. If is a Lie -algebroid then is a Lie -algebroid.