I am wondering whether the concept of the flow of a vector field can be described in the following sheaf-theoretic way: $\newcommand{\Flow}{\mathrm{Flow}} \newcommand{\pre}{\mathrm{pre}} \renewcommand{\epsilon}{\varepsilon} \renewcommand{\phi}{\varphi} \newcommand{\from}{\colon} \newcommand{\blank}{{-}}$
Let $M$ be a smooth manifold. Let me define the presheaf $\Flow^\pre$ on $M$ given as follows: If $U \subset M$ is an open subset, then an element of $\Flow^\pre(U)$ should be given by a pair $(\epsilon, \phi)$ where $\epsilon$ is some positive real number and $\phi \from U \times (-\epsilon, \epsilon) \to M$ is a smooth map satisfying some properties, modulo some equivalence relation. The properties should be:
- $\phi(x,0) = x$ for all $x \in U$.
- $\phi( \phi(x,t), s) = \phi(x, t+s)$ whenever $s,t \in (-\epsilon,\epsilon)$, $x \in U$ such that $t + s \in (-\epsilon, \epsilon)$ and $\phi(x,t) \in U$.
The equivalence relation should be that $(\epsilon, \phi) \sim (\epsilon', \phi')$ if there exists some $\epsilon'' \le \min(\epsilon, \epsilon')$ such that $\phi(x,t) = \phi'(x,t)$ for all $x \in U$ and $t \le \epsilon''$.
I think that the above defines a separated presheaf on $M$. There is a morphism of presheaves from $\Flow^\pre$ to the sheaf of smooth vector fields on $M$. If $U \subset M$ is an open subset and $[\epsilon, \phi] \in \Flow^\pre(U)$ then a vector field $X$ is given by stipulating that $X_x = d_0\phi(x, \blank) (\partial/\partial t)$. Here $d_0\phi(x,\blank)$ denotes the differential at $0$ of the smooth map $\phi(x, \blank) \from (-\epsilon, \epsilon) \to M$.
Denote by $\Flow$ the sheafification of $\Flow^\pre$. By the universal property of sheafification, there is an induced morphism of sheaves from $\Flow$ to the sheaf of smooth vector fields. I think that the usual construction of a flow of a vector field (proved using the Cauchy-Lipschitz theorem) says that this morphism is an isomorphism of sheaves. Is that correct?