[Garlef]

> The whole thing is just about arrow composition!

There's much much more to it.

For example, a version of the yoneda lemma also holds for metric spaces (instead of a set of arrows between to things, you simply have a number indicating a distance between two things).

Here's how I like to think about the yoneda lemma:

If you have some kind of objects you want to talk about, one way to do this is by relating these objects to each other. Once you have established such a "method of discourse" (i.e. a way to talk about how your objects relate to each other) the yoneda lemma tells you:

1. You can forget about the inner structure of your objects; Everything is already contained in your "method of discourse".

2. Inversely: The choice of a "method of discourse" severely limits what you can say about your objects.

3. There is no spoon: To understand how you can escape these limitations you need a "method of discourse" for "methods of discourse".

[myWindoonn]

I like your example and perspective. I would just add on that it is not facile to do this for metric spaces, but leads directly to constructive solid geometry [0], where we render images of complex solid objects by exchanging the object for a signed distance function [1], a function which indicates how far the object is from any point in the space.

[0] https://en.wikipedia.org/wiki/Constructive_solid_geometry

[1] https://en.wikipedia.org/wiki/Signed_distance_function

[a1369209993]

> a number indicating a distance between two things

That *is* a arrow. Arrow composition is adding up the distance along a path.

[Garlef]

The correct term would be "composition in an enriched category". The morphism objects in this context are usually not called arrows.

I think the other poster meant "elements in the set of morphisms" when they said "arrows".

The difference is the following: Metric spaces "are" categories enriched over the real numbers, ordinary categories are categories enriched over the category of sets. So in one case the morphism objects are sets while in the other case they are real numbers. "Arrows" then refers to something internal to the morphism object. The distance (a real number) between two points in a metric space does not have any internal structure.

[AnimalMuppet]

That almost describes discourse about politics and social issues, too.