Quantization and Cohomology (Week 21)
Posted by John Baez
This week in our course on Quantization and Cohomology we used Chen’s ‘smooth space’ technology to implement a new approach to Lagrangian mechanics, based on a smooth category equipped with an ‘action’ functor:

Week 21 (Apr. 17)  Any quotient of a smooth space becomes a
smooth space. The category of smooth spaces has pushouts.
The category of smooth spaces is cartesian closed. The path groupoid $P X$ of a smooth space $X$. The path groupoid is a smooth category. Smooth functors.
Theorem: a smooth functor $S: P X \to \mathbb{R}$ is the same as a 1form
on X.
Supplementary reading:

John Baez and Urs Schreiber,
Higher gauge theory II: 2connections, draft version.
Section 6.1: proof that for any Lie group $G$, smooth functors $S: P X \to G$ are the same as $Lie (G)$valued 1forms on $X$

John Baez and Urs Schreiber,
Higher gauge theory II: 2connections, draft version.
Last week’s notes are here; next week’s notes are here.
In a bit more detail: we saw that any smooth space $X$ has a smooth groupoid of paths $P X$. The Lagrangian approach to classical mechanics involves a smooth groupoid $C$ where the objects are ‘configurations’ of our system and the morphisms are ‘processes’ or ‘paths’. The action should define a functor $S: C \to \mathbb{R}$. So, it’s nice that in the special case when $C$ is a path groupoid, such functors turn out to be familiar entities! They’re just 1forms on $X$.
However, this isn’t quite general enough. What we really want is something that looks locally like a 1form on $X$, but not globally: a connection on a $U(1)$ bundle over $X$! This, after all, is what people use in geometric quantization — usually in the special case where $X$ is a symplectic manifold.
To get this answer, we’ll need to generalize from smooth functors to smooth anafunctors, as defined by Toby Bartels. A smooth anafunctor is something that’s locally isomorphic to a smooth functor!