# nLab topos of laws of motion

Contents

topos theory

## Theorems

#### Synthetic differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

$\array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }$

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

# Contents

## 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.

The notion originates around

and was made more explicit in