Groupoid morphism from time pair to phase space - what is it? I was reading the answer to the this post on physics.stack and at the very begining the answer considers a nonautonomous ODE:
$$\frac{dX}{dt} = f(t,X)$$
and says that it generates a flow, which is a groupoid morphism from time pairs to diffeomorphisms of phase space. I have no background in dynamical system, so could someone please clarify to me what these terms groupoid morphism and diffeomorphism of phase space actually mean and why does the above equation generate such objects? If this is too difficult to explain in a math.stack post, maybe you can point me out a good reference?
 A: This isn't about some deep background, but basic definitions.
We have the category/groupoid of time: The objects are just individual moments of time, parametrized by $\mathbb R$. There is a single morphism from any $t_1$ to $t_2$ ($t_2<t_1$ is allowed), so it forms a groupoid. In other words, there is a bijection between pair of numbers and morphisms.
We also have the category/groupoid of phase spaces: There is a single object which is the manifold of phases (e.g. the cotangent bundle of the configuration manifold), and the morphisms are diffeomorphisms.
Given a dynamical system like $X'(t) = f(t,X)$, for any initial data $X(t_1) = X_1$, under suitable continuity and smoothness assumptions, it defines a unique solution $X(t)$, then for each $t_2$, we get $X_{t_1, X_1}(t_2)$, therefore $X(t_2)$ is in the diffeomorphism group of the phase space if we fix $t_1, t_2$ and let $X_1$ vary.
P.S. I have a slight issue with this: the flow $X_{t_1, X_1}(t)$ might blow up in finite time, so the flow is not necessarily universally defined... but I guess this is not the point of that post. I guess it's reasonable to assume there is no finite time singularity in any realistic physical model, unless the model expires under harsher conditions.
