Here is a general observation which occured to me this morning. It’s a half-baked thought, but maybe somebody can see something worthwhile in it.

First, I have to apologize for writing “”configuration space” in my above comment # a couple of times when I really should have said “target space”. This is the worst kind of mistake one can make in this kind of discussion, where everything is about the subtle differences between all these concepts, and I am feeling really bad about it. I have now corrected that in the above comment.

So what really happened this morning was that I noticed this recurring misprint. But as it goes with mistakes, they sometimes lead us to deeper insights.

So let me amplify the *correct* statement once more:

configuration space is the space of maps from parameter space into target space
$\mathrm{conf} = [\mathrm{par},\mathrm{tar}]$
(or a subthing thereof, but let’s not worry about such details at the moment).

history space is the space of maps from the worldvolume into target space
$\mathrm{hist} = [\mathrm{worldvol},\mathrm{tar}]$

Now, what people like Rovelli call the “extended configuration space” is the *product* of these two spaces
$\mathrm{extconf} =
\mathrm{worldvol}\times\mathrm{tar}
\,.$
Moreover, it turns out that *sheaves* (of algebras of observables) on the extended configuration space play an important role.

Staring at these facts for a moment seems to suggest that this is telling us that we should pass from functors
$\mathrm{hist} \ni \gamma :
\mathrm{worldvol} \to \mathrm{tar}$
to the corresponding profunctors
$\tilde \gamma : \mathrm{worldvol}^{\mathrm{op}}\times
\mathrm{tar}
\to
\mathrm{Set}
\,.$

These now can loosely be thought of as *set-valued functions on the extended configuration space*. That might be remarkable, given that we noticed that some things in quantum theory might become much clearer would we be dealing with set-valued functions instead of with ordinary functions that take values in numbers.

Maybe even better, this might be a way to see *why* we should in fact consider not just sets, but $\mathbb{C}$-sets:

because what are these sets that $\tilde \gamma$ associates to a point in the extended configuration space? Well, these are the *collection of morphisms* to that point from a given point, of course, but here this means that these are the *set of paths* between these two points.

So given some functor
$\mathrm{tar} \to \mathrm{phas}$
that associates phases to paths, we can naturally compose it with our $\tilde \gamma$ in some way – and this will label all elements of these sets with their associated phase.

So, given such an action functor, we naturally get a profunctor
$(\tilde \gamma,\mathrm{tra})
:
\mathrm{extconf} \to \mathbb{C}\mathrm{Set}
\,.$

The process of path integration would then be the postcomposition of this with the cardinality operation.

Hm…

## Re: Quantization and Cohomology (Week 17)

It is interesting how you set up the inner product in this week’s notes.

After declaring your Hilbert spaces to be spaces of functions on configuration space $X$, you don’t set $\langle \phi | \psi \rangle = \int_{X} \bar \phi(x) \psi(x)\; dx$ as the naive reader might have expected you would. Instead, you do something more like a correlator and set $\langle \phi | \psi \rangle = \int_{x \stackrel{\gamma}{\to} y} \bar \phi(x) \psi(y) \; e^{i S(\gamma)}d\mu \,.$

What precisely this will mean will in particular depend on the precise nature of paths that are integrated over.

I think I know where you are heading: constrained dynamics. Your way to look at the inner product will be more immediately applicable to the relativistic than to the non-relativistic particle, for instance.

And that’s also why you base category $C$ looks like paths in $Q \times \mathbb{R}$, instead of like paths in $Q$, for a particle propagating on

space$Q$.I need to think more about this point of view.

Differing approaches to this issue might, for instance, affect discussions like I had with Jeffrey, over on the canonical measure thread (his comment, my reply).