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\begin{document}

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\section*{shifted tangent bundle}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{supergeometry}{}\paragraph*{{Super-Geometry}}\label{supergeometry}

[[!include supergeometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{alternative_characterizations}{Alternative characterizations}\dotfill \pageref*{alternative_characterizations} \linebreak
\noindent\hyperlink{as_an_nqsupermanifold}{As an NQ-supermanifold}\dotfill \pageref*{as_an_nqsupermanifold} \linebreak
\noindent\hyperlink{as_the_tangent_lie_algebroid}{As the tangent Lie algebroid}\dotfill \pageref*{as_the_tangent_lie_algebroid} \linebreak
\noindent\hyperlink{as_an_internal_hom_object}{As an internal hom object}\dotfill \pageref*{as_an_internal_hom_object} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

The \emph{shifted tangent bundle} or \emph{odd tangent bundle} $\Pi T X$ 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.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

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

\begin{displaymath}
C^\infty(\Pi T X) := \Omega^\bullet(X)
  = \wedge^\bullet_{C^\infty(X)} \Gamma(T^* X)
  \,.
\end{displaymath}
More precisely, for each open $U \subset X$ the value of the [[structure sheaf]] of $\Pi T X$ is $\Omega^\bullet(U)$.

For more details see at \emph{[[geometry of physics -- supergeometry]]}.

\hypertarget{alternative_characterizations}{}\subsection*{{Alternative characterizations}}\label{alternative_characterizations}

\hypertarget{as_an_nqsupermanifold}{}\subsubsection*{{As an NQ-supermanifold}}\label{as_an_nqsupermanifold}

In the context of [[supergeometry]] the algebra $\Omega^\bullet(X)$ 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 $\Pi T X$, which as an odd derivation on the function algebra $\Omega^\bullet(X)$ is just the [[deRham complex|deRham differential]].

Equipped with this structure $\Pi T X$ is naturally an [[NQ-supermanifold]].

\hypertarget{as_the_tangent_lie_algebroid}{}\subsubsection*{{As the tangent Lie algebroid}}\label{as_the_tangent_lie_algebroid}

The [[dg-algebra]] $(\Omega^\bullet(X), d_{dR})$ 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 $\Pi T X$ corresponds to the fact that these are the tangents to the [[k-morphism|1-morphism]]s 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).

\hypertarget{as_an_internal_hom_object}{}\subsubsection*{{As an internal hom object}}\label{as_an_internal_hom_object}

With $\mathbb{R}^{0|1}$ denoting the [[odd line]], i.e. the [[supermanifold]] with function algebra the algebra of [[dual number]]s, one finds that

\begin{displaymath}
\Pi T X = [\mathbb{R}^{0|1}, X]
\end{displaymath}
is the [[internal hom]] object in the category of [[supermanifold]]s of maps from $\mathbb{R}^{0|1}$ 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 [[representable functor|represented by]] $\mathbb{R}^{0|1}$ and $X$, is itself [[representable functor|representable]] and is represented by $\Pi T X$.

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

\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[geometry of physics -- supergeometry]] the section \emph{\href{geometry+of+physics+--+supergeometry#SuperMappingSpaces}{Super mapping spaces}}

\end{itemize}
[[!redirects shifted tangent bundles]]

[[!redirects odd tangent bundle]] [[!redirects odd tangent bundles]]



\end{document}