"Lie theory" for anchored bundles and reflexive graphs Perhaps Lie theory is not the correct term, but I'm thinking of the intermediate result in the Lie groupoid to Lie algebroid correspondence. Given a Lie groupoid $G$ over $M$, we may construct the Lie algebroid of $G$ by taking the pullback of the cospan
$$ M \xrightarrow{(e,0)} G \times TM \xleftarrow{(p, Ts)} TG$$
Now, if $G$ were just a reflexive, involutive graph rather than a groupoid, we would still get a vector bundle over $M$ from this pullback, with an anchor $\varrho := A \hookrightarrow TG \xrightarrow{Tt} TM$.
Has anyone written down the analogue of "Lie integration" for this setting? If I move to synthetic differential geometry there's a way through using enriched sketches, but I'd like to see something a little more down to earth.
 A: Given a vector bundle $E\to M$ equipped with an anchor map $\rho:E\to TM$, choose a connection $\nabla$ on $E$ and consider the ODE
\begin{align*}
\dot\gamma(t) &= \rho(v(t))\\
(\gamma^*\nabla)_t v(t) &= 0
\end{align*}
for $\gamma:I\to M$ a path in $M$ and $v\in\Gamma(\gamma^*E)$ a section of the pulled back vector bundle. Standard results imply that for $(x_0,v_0)\in E$, there is an open interval $I(x,v)$ containing $0$ such that there is a unique solution $(x,v)$ of this ODE on $I$ with $(x(0),v(0)) = (x_0,v_0)$ and that $I(x,v)$ is maximal with respect to this property.
Let $U\subset E$ be the (open) subset of elements such that $1\in I(x,v)$, and note that it contains the zero section since constant paths with $v\equiv 0$ define a global solution of the ODE; then we have an obvious bijection between $U$ and germs of solutions $(x(t),v(t))$ of the ODE on $[0,1]$. In particular, the involution $(x(t),v(t))\mapsto (x(1-t),-v(1-t))$ on the latter space gives rise to an involution $\Phi:U\cong U$ fixing the zero section, and standard results show that it is smooth (it is essentially given by sending an element $(x,v)$ to the time $1$ value of the ``flow'' generated by it).
We obtain a reflexive, involutive graph by setting

*

*$G = U$

*$s = p:G\to M$ the bundle projection

*$t = p\circ \Phi:G\to M$

*$e:M\to G$ the inclusion of the zero section

*Lastly, the involution is given by $i = \Phi:U\cong U$.

It is straightforward, if tedious, to verify that applying your construction to this data gives back $E\xrightarrow{\pm \rho}M$ (the sign depends on some conventions and can be absorbed into the definition of the ODE).
This construction is essentially given by integrating the $L_\infty$-algebroid generated by $E\xrightarrow{\rho}M$ to an $L_\infty$-groupoid $\Delta^{op}\to \operatorname{Man}$ and restricting to $\Delta_{\le 1}^{op}$.
