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\begin{document}

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\section*{synthetic tangent bundle}

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

\hypertarget{synthetic_differential_geometry}{}\paragraph*{{Synthetic differential geometry}}\label{synthetic_differential_geometry}

[[!include synthetic differential geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{for_microlinear_spaces}{For Microlinear spaces}\dotfill \pageref*{for_microlinear_spaces} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

In [[synthetic differential geometry]], the \emph{[[tangent bundle]]} of an [[object]] $X$ is the [[internal hom]] $X^{\mathbb{D}^1}$ out of the 1d first order [[infinitesimal disk]] $\mathbb{D}^1$, equipped with the projection to $X$ induced from the unique point $\ast \to \mathbb{D}^1$:

\begin{displaymath}
X^{(\ast \to \mathbb{D}^1)} \;\colon\; X^{(\mathbb{D}^1)} \longrightarrow X
  \,.
\end{displaymath}
(Here we are using that the [[internal hom|internal]] [[hom-functor]] $(-)^{(-)}$ is a [[contravariant functor]] in its ``exponent'' variable.)

This makes manifest and precise the intuitive idea that a [[tangent vector]] on $X$ is an ``[[infinitesimal object|infinitesimal]] [[curve]]'' in $X$, see also the \hyperlink{Examples}{Examples} below.

In this formulation the operation of [[differentiation]] is simply the [[internal hom]]-[[functor]]:

Given a function:

\begin{displaymath}
X \overset{f}{\longrightarrow} Y
  \,.
\end{displaymath}
its [[differential]] is its image under the [[internal hom]] $(-)^{(\mathbb{D}^1)}$:

\begin{displaymath}
X^{(\mathbb{D}^1)}
   \overset{ \phantom{AA} f^{(\mathbb{D}^1)} \phantom{AA} }{\longrightarrow}
  Y^{(\mathbb{D}^1)}
  \,.
\end{displaymath}
\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

In a standard model for [[synthetic differential geometry]]/[[differential cohesion]] such as the [[Cahiers topos]] $\mathbf{H}$, for $X \in SmthMfd \overset{y}{\hookrightarrow} \mathbf{H}$ an ordinary [[smooth manifold]], its [[synthetic tangent bundle]] coincides with the traditional [[tangent bundle]] $T X \overset{p}{\to} T X$:

\begin{displaymath}
\itexarray{
    X^{(\mathbb{D}^1)} & \simeq & T X
    \\
    {}^{\mathllap{ X^{ (\ast \to \mathbb{D}^1) } }}\big\downarrow 
      && 
    \big\downarrow{}^{ \mathrlap{p_X} }
    \\
    X^{\ast} & \simeq & X
  }
\end{displaymath}
Moreover, if $Y \in SmthMfd \hookrightarrow \mathbf{H}$ is another [[smooth manifold]], then a [[morphism]]

\begin{displaymath}
X \overset{\phantom{AA} f \phantom{AA} }{\longrightarrow} Y
\end{displaymath}
is equivalently a [[smooth function]] in the traditional sense (i.e. the external [[Yoneda embedding]]-[[functor]] $SmthMfd \overset{y}{\hookrightarrow} \mathbf{H}$ is [[fully faithful functor|fully faithful]] ) and its [[image]] under the [[internal hom]] is its traditional [[differentiation]] $d f$:

\begin{displaymath}
\itexarray{
    X^{(\mathbb{D}^1)} 
      &\overset{ \phantom{AA} f^{(\mathbb{D}^1)} \phantom{AA} }{\longrightarrow}&
    Y^{(\mathbb{D}^1)}
    \\
    {}^\simeq\big\downarrow && \big\downarrow{}^\simeq
    \\
    T X &\overset{ \phantom{AA} d f \phantom{AA} }{\longrightarrow}& 
    T Y
  }
\end{displaymath}
This way the evident [[functor|functoriality]] of the [[internal hom]] $(-)^{(\mathbb{D}^1)}$ is identified with the [[chain rule]] of traditional [[differentiation]].

For more on this see

\begin{itemize}%
\item at \emph{\href{Cahiers+topos#RelationToSyntheticTangentSpaces}{Cahiers topos -- Synthetic tangent spaces}}.


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



\end{itemize}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{for_microlinear_spaces}{}\subsubsection*{{For Microlinear spaces}}\label{for_microlinear_spaces}

For $X$ a [[microlinear space]] the synthetic tangent bundle shares many of the expected properties of a [[tangent bundle]].

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[odd tangent bundle]] (analog in [[supergeometry]])


\item [[kinematic tangent bundle]], [[operational tangent bundle]]


\item [[jet bundle]] via [[jet comonad]]



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

For lecture notes see

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

\end{itemize}
[[!redirects synthetic tangent bundles]] [[!redirects synthetic tangent space]] [[!redirects synthetic tangent spaces]]



\end{document}