The n-Category Café

Yes. I often try to explain the Yoneda Lemma by saying an object is determined by the structure of all the morphisms to it… or more vaguely, a thing is known by its relation to other things. It’s great to see Barry Mazur give a very similar, but more well-worked out, description.

I recently found this 1998 quote from the computing pioneer, Alan Kay:

Just a gentle reminder that I took some pains at the last OOPSLA to try to remind everyone that Smalltalk is not only NOT its syntax or the class library, it is not even about classes. I’m sorry that I long ago coined the term “objects” for this topic because it gets many people to focus on the lesser idea.

The big idea is “messaging” - that is what the kernel of Smalltalk/Squeak is all about (and it’s something that was never quite completed in our Xerox PARC phase). The Japanese have a small word - ma - for “that which is in between” - perhaps the nearest English equivalent is “interstitial”. The key in making great and growable systems is much more to design how its modules communicate rather than what their internal properties and behaviors should be. Think of the internet - to live, it (a) has to allow many different kinds of ideas and realizations that are beyond any single standard and (b) to allow varying degrees of safe interoperability between these ideas.

That feels right to me: like the Yoneda Lemma, Mach’s principle—taken to an extreme—claims that the properties of each object are determined by its relationship to all other objects.

While Einstein was inspired by Mach’s principle when formulating general relativity, the principle was never made very precise, so it’s a bit unclear whether general relativity obeys Mach’s principle. There are arguments that it does not. From Wikipedia:

Mach’s principle says that this is not a coincidence—that there is a physical law that relates the motion of the distant stars to the local inertial frame. If you see all the stars whirling around you, Mach suggests that there is some physical law which would make it so you would feel a centrifugal force. There are a number of rival formulations of the principle. It is often stated in vague ways, like “mass out there influences inertia here”. A very general statement of Mach’s principle is “Local physical laws are determined by the large-scale structure of the universe.”

This concept was a guiding factor in Einstein’s development of the general theory of relativity. Einstein realized that the overall distribution of matter would determine the metric tensor, which tells you which frame is rotationally stationary. Frame dragging and conservation of gravitational angular momentum makes this into a true statement in the general theory in certain solutions. But because the principle is so vague, many distinct statements can be (and have been) made which would qualify as a Mach principle, and some of these are false. The Gödel rotating universe is a solution of the field equations which is designed to disobey Mach’s principle in the worst possible way. In this example, the distant stars seem to be revolving faster and faster as one moves further away. This example doesn’t completely settle the question, because it has closed timelike curves.

On the other hand, ‘frame-dragging’ is often cited in support of the claim that Mach’s principle holds in general relativity: if you are inside by a large massive sphere that’s spinning around, you’ll feel a force similar to the ‘centrifugal force’ you’d feel if the sphere were stationary but you were spinning around!

In short, I think Mach’s principle would need to be made more precise before we could decide to adopt it or compare it to the Yoneda Lemma. Ideally the Yoneda Lemma would be a way to make Mach’s principle more precise, but this would require a specific kind of category-theoretic reformulation of physics.