The shifted tangent bundle or odd tangent bundle of a manifold is an incarnation of the ordinary tangent bundle of as a supermanifold with the underlying manifold in even degree and the tangent vectors in odd degree.
More precisely, for each open the value of the structure sheaf of is .
For more details see at geometry of physics -- supergeometry.
In the context of supergeometry the algebra is regarded as a -graded algebra, but of course this -grading lifts to an -grading in the obvious way.
Moreover, there is a canonical odd vector field on , which as an odd derivation on the function algebra is just the deRham differential.
Equipped with this structure is naturally an NQ-supermanifold.
The dg-algebra may also be regarded as the Chevalley-Eilenberg algebra of the tangent Lie algebroid of , which identifies the shifted tangent bundle in its refinement to an NQ-supermanifold with the tangent Lie algebroid of .
From this perspective, the fact that the vectors are regarded as being in degree one in corresponds to the fact that these are the tangents to the 1-morphisms of the fundamental groupoid of . (Which is denoted but with the “” of completely different meaning than the “” as used here, which just indicates degree shift).
is the internal hom object in the category of supermanifolds of maps from to . More precisely this means that the internal hom which exists in the closed monoidal structure on presheaves on the category of supermanifolds, between the presheaves represented by and , is itself representable and is represented by .
The existence of the structure of an NQ-supermanifold is from this point of view a consequence of the fact that naturally carries an action of the endomorphism object . For more on this see NQ-supermanifold.