nLab
shifted tangent bundle

Idea

The shifted tangent bundle $\Pi TX$ of a manifold $X$ is an incarnation of the ordinary tangent bundle of $X$ as a supermanifold with the underlying manifold in even degree and the tangent vectors in odd degree.

Definition

For $X$ a smooth manifold the supermanifold $\Pi TX$ is defined to be the one specified by the fact that its superalgebra of functions is the graded-commutative exterior algebra of differential forms

$$C^{\mathrm{\infty }}(\Pi TX):={\Omega }^{•}(X)={\wedge }_{C^{\mathrm{\infty }}(X)}^{•}\Gamma (T^{*}X).$$C^\infty(\Pi T X) := \Omega^\bullet(X) = \wedge^\bullet_{C^\infty(X)} \Gamma(T^* X) \,.

More precisely, for each open $U\subset X$ the value of the structure sheaf of $\Pi TX$ is ${\Omega }^{•}(U)$.

Alternative characterizations

As an NQ-supermanifold

In the context of supergeometry the algebra ${\Omega }^{•}(X)$ is regarded as a ${\mathbb{Z}}_2$-graded algebra, but of course this ${\mathbb{Z}}_2$-grading lifts to an $\mathbb{N}$-grading in the obvious way.

Moreover, there is a canonical odd vector field $v_d$ on $\Pi TX$, which as an odd derivation on the function algebra ${\Omega }^{•}(X)$ is just the deRham differential.

Equipped with this structure $\Pi TX$ is naturally an NQ-supermanifold.

As the tangent Lie algebroid

The dg-algebra $({\Omega }^{•}(X),d_{\mathrm{dR}})$ may also be regarded as the Chevalley-Eilenberg algebra of the tangent Lie algebroid of $X$, which identifies the shifted tangent bundle in its refinement to an NQ-supermanifold with the tangent Lie algebroid of $X$.

From this perspective, the fact that the vectors are regarded as being in degree one in $\Pi TX$ corresponds to the fact that these are the tangents to the 1-morphisms of the fundamental groupoid of $X$. (Which is denoted $\Pi (X)$ but with the “$\Pi$” of completely different meaning than the “$\Pi$” as used here, which just indicates degree shift).

As an internal hom object

With ${ℝ}^{0\mid 1}$ denoting the odd line, i.e. the supermanifold with function algebra the algebra of dual numbers, one finds that

$$\Pi TX=[{ℝ}^{0\mid 1},X]$$\Pi T X = [\mathbb{R}^{0|1}, X]

is the internal hom object in the category of supermanifolds of maps from ${ℝ}^{0\mid 1}$ to $X$. 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 ${ℝ}^{0\mid 1}$ and $X$, is itself representable and is represented by $\Pi TX$.

The existence of the structure of an NQ-supermanifold is from this point of view a consequence of the fact that $[{ℝ}^{0\mid 1},X]$ naturally carries an action of the endomorphism object $\mathrm{End}({ℝ}^{0\mid 1})$. For more on this see NQ-supermanifold.