Let's Grothendieck Dynamical Systems

In week 3 we described the fundamental groupoid functor

$ \pi_1 \colon \mathsf{Top} \to \mathsf{Grpd}$

which sends a topological space $ X$ to the groupoid where objects are points in $ X$ and morphisms are equivalence classes of paths.

There is a similar story for directed topological spaces; spaces equipped with a set of paths closed under concatenation, subpaths, and constant paths. Each directed topological space generates a fundamental category where the objects are the points of your directed topological space and the morphisms are given by the set of paths. In particular, every dynamical system gives a directed topological space. I won't describe the fundamental category functor in full generality but instead describe it as a functor from the category of dynamical systems.

Last time we constructed the category $ \mathsf{Dynam}$ of dynamical systems as the $ \mathsf{Diff}$-enriched functor category

$ \mathsf{Diff}^{\mathbb{R}}$

where $ \mathsf{Diff}$ is the category of diffeological spaces and $ \mathbb{R}$ is the $ \mathsf{Diff}$ enriched category with one object denoted by $ \mathbf{\cdot}$ and every real number as a morphism. If you're scared by the enrichment it might help to note that $ \mathsf{Diff}$-functors are standard functors except that the morphisms are mapped in a smooth way and $ \mathsf{Diff}$-natural transformations are just ordinary natural transformations.

$ \uparrow \Pi_1 \colon \mathsf{Dynam} \to \mathsf{Cat}$

is the functor which sends a dynamical system $ \phi \colon \mathbb{R} \to \mathsf{Diff}$ to the category where

• objects are given by the points of $ \phi(\mathbf{\cdot})$ i.e. the points of the manifold which the dynamical systems lives and,
• there is a morphism $ t\colon x \to y$ if the flow of $ \phi$ sends $ x$ to $ y$ after time $ t$. That is there exists some $ t \in \mathbb{R}$ such that $ \phi(t)(x) = y$.
• Composition is given by concatenation. For a morphism $ t \colon x \to y$ and a morphism $ t' \colon y \to z$ there is a morphism $ t' \circ t \colon x \to z$ because $ \phi(t+t')(x) = \phi(t')(\phi(t,x)) = \phi(t', y) = z$.
• The identity morphism for $ x$ is given by $ 0 \colon x \to x$ because $ \phi(0,x) = x$.

For a morphism of dynamical systems i.e. a natural transformation $ \alpha \colon \phi \Rightarrow \psi$ there is a functor

$ \uparrow \Pi_1 (\alpha) \colon \uparrow \Pi_1 (\phi) \to \uparrow \Pi_1 (\psi)$

which sends

• an object $ x \in \uparrow \Pi_1 (\phi)$ to the object $ \alpha_{\mathbf{\cdot}} (x) \in \uparrow \Pi_1 (\psi)$ and,
• a morphism $ t \colon x \to y \in \uparrow \Pi_1 (\phi)$ to the morphism $ t \colon \alpha_{\mathbf{\cdot}} (x) \to \alpha_{\mathbf{\cdot}} (y)$. This is well defined because $ \alpha$ is a natural transformation.

You may notice that this is more than a category, it's a groupoid because every morphism $ t \colon x \to y$ has an inverse $ -t \colon y \to x$. It may seem odd to you to call this the fundamental category functor. The reason for this is that it is a restriction of a more general functor which maps directed topological spaces to categories with not-neccesarily invertible morphisms.

What happens next is in the title.

The Grothendieck construction of $ \uparrow \Pi_1  \colon \mathsf{Dynam} \to \mathsf{Cat}$ is a category $ \int \uparrow \Pi_1$ where

• objects are pairs $ (\phi, x)$ where $ \phi \colon \mathbb{R} \to \mathsf{Diff}$ is a dynamical system and $ x$ is a point in its manifold.
• A morphism from $ (\phi, x)$ to $ (\psi, y)$ is a tuple $ (\alpha \colon \phi \Rightarrow \psi, t \colon \alpha_{\cdot} (x) \to y)$ where $ \alpha$ is a morhpism of dynamical systems and $ t$ is a morphism in $ \uparrow \Pi_1 (\psi)$. In words, a morphism from $ (\phi,x)$ to $ (\psi,y)$ is a morphism of dynamical systems along with a flow $ t$ from the image of $ x$ to $ y$.

There are two interesting interpretations of this category. In week 3 we performed a similar Grothendieck construction of the fundamental groupoid functor. This gave a category where the objects are pointed topological spaces and the morphisms are continuous maps which preserve the basepoint up to an invertible path.  The category $ \int \uparrow \Pi_1$ has a similar interpretation as pointed dynamical systems. The morphisms in $ \int \uparrow \Pi_1$ are morphisms of dynamical systems which preserve the basepoint up to a flow.

The second interpretation is as the category of solutions to an intial value problems. An object $ (\phi,x) \in \int \uparrow \Pi_1$ is a dynamical system equipped with an initial condition. The flow of $ \phi$ starting from this intial condition gives the solution to your differential equation with initial condition $ x$. A morphism $ (\alpha, t)$ can be interpreted physically as well.  This is a smooth map between the underlying manifolds such that flow of $ \phi$ starting at $ x$ matches the flow of $ \psi$ starting at $ y$ after some time $ t$.