nLab topos of laws of motion

Idea

In (Lawvere 97) it was observed that equations of motion in physics can (almost, see below) be formalized in synthetic differential geometry as follows.

Let $\mathbf{H}$ be an ambient synthetic differential topos (such as the Cahiers topos of smooth spaces and formal smooth manifolds).

The canonical line object $\mathbb{A}^1 = \mathbb{R}$ of this models the continuum line, the abstract worldline. Let

$$D \hookrightarrow \mathbb{R}$$

be the inclusion of the first order infinitesimal neighbourhood of the origin of $\mathbb{R}$ – in the internal logic this is $D = \{x \in \mathbb{R}| x^2 = 0\}$, externally it is the spectrum of the ring of dual numbers over $\mathbb{R}$.

Then for $X \in \mathbf{H}$ any object which we are going to think of as a configuration space of a physical system. For instance if the system is a particle propagating on a spacetime, then $X$ is that spacetime. Or $X$ may be the phase space of the system.

Accordingly the mapping space $[\mathbb{R}, X] \in \mathbf{H}$ is the smooth path space of $X$. This is the space of potential trajectories of the physical system.

If $X$ is thought of as phase space, then every point in there determines a unique trajectory starting at that point. This means that time evolution is then an action of $\mathbb{R}$ on $X$. As $X$ here might be any space, we have the collection

$$\mathbb{R}Act(\mathbf{H}) \in Topos$$

of all $\mathbb{R}$-actions on objects in $\mathbf{H}$. This is again a topos, and hence this is a first version of what one might call a topos of laws of motion.

On the other hand, if we think of $X$ as configuration space, then it is (in the simplest but common case of physical systems) a tangent vector in $X$ that determines a trajectory, hence a point in $[D,X]$. There is the canonical projection $[\mathbb{R},X] \longrightarrow [D,X]$ from the smooth path space to the tangent bundle, which sends each path to its tangent vector/derivative at $0 \in \mathbb{R}$. A section of this map is hence an assignment that sends each tangent vector to a trajectory which starts out with this tangent. Specifying such a section is hence part of what it means to have equations of motion in physics. Accordingly in Toposes of laws of motion Lawvere called the collection of such data a Galilian topos of laws of motion.

Of course this is not quite yet what is actually used and needed in physics. On p. 9 of (Lawvere 97) this problem is briefly mentioned:

But what about actual dynamical systems in the spirit of Galileo, for example, second-order ODE’s? (Of course, the symplectic or Hamiltonian systems that are also much studied do address this question of states of Becoming versus locations of Being, but in a special way which it may not be possible to construe as a topos;

It turns out that it does exist as a “higher topos”.

In Classical field theory via Cohesive homotopy types it is observed that if $\mathbf{H}$ to be not just a topos but an infnity-topos, then genuine classical mechanics governed by Hamilton's equations and Hamilton-Jacobi theory arises by actions of $\mathbb{R}$ on objects in the slice (infinity,1)-topos $\mathbf{H}_{/\mathbf{B}U(1)_{conn}}$, where $\mathbf{B}U(1)_{conn} \in \mathbf{H}$ is the moduli stack of circle group-principal connections. So

$$\mathbb{R}Act\left( \mathbf{H}_{/\mathbf{B}U(1)_{conn}} \right)$$

is actually a “topos of laws of motion” in the sense of Hamilton-Lagrange-Jacobi classical mechanics. For more on this see Higher toposes of laws of motion.

Related concepts

• Higher toposes of laws of motion

References

The notion originates around

• William LawvereCategorical dynamics, 1967 Chicago lectures (pdf)

and was made more explicit in

• William Lawvere, Toposes of laws of motion, 1997