> I'm curious why the video refers to the surface of the manifold - I
> would have thought that cats would be inside the manifold as well.

Not necessarily.  The surface of the manifold is defined as the
collection of points that describe cats.  There's no guarantee that the
surface forms a closed volume.  We chose to depict it that way, but it
doesn't have to be.  If it's not completely closed, then there is no
"inside".

Another way to think about it: if we require the manifold to be a closed
surface and say that any point within that volume describes a cat, then
for any two nearby points within the volume, we could linearly
interpolate between them and the result would also be a cat.  This is a
very strong assumption and almost certainly does not hold in real cat
manifolds.  In contrast, saying that all cat points lie on the SURFACE
is much weaker; it only implies that there is SOME path between them
that lies on the surface, not that a linear path does.

-- Dave