Yoneda Lemma

Why is it interesting?

The Yoneda lemma is "arguably the most important result of category theory" (source).

It is important for physics, because it allows us to make the ideas of relational physics precise.

Layman

Informally, [the Yoneda Lemma] implies that you can gain information about an object by viewing it from *all* possible angles. That kinda makes sense, right? To put it another way: a mathematical object is totally determined by its relationships to other objects. It's sort of like the old saying, “Tell me who your friends are, and I’ll tell you who *you* are.”Math3ma

Student

[T]he collection of all ways to relate an object to other objects is isomorphic to the object itself.https://news.ycombinator.com/item?id=7715277

[A]n object is completely determined by its relationships to other objects. http://www.math3ma.com/mathema/2017/9/14/the-yoneda-lemma

One way to think of the Yoneda lemma is precisely this: that the objects of any category can be interpreted as sets with extra structure. Think about this a minute. We have an abstract category C and we wish to associate to each object of C some set equipped with extra structure. Moreover, we want to do this in a way which completely records everything there is to know about this object. How can we do it?

Well, the only interesting thing about an object in a category is its morphisms to and from other objects, and how these compose with *other* morphisms. This principle should be our guide.

So, what should we do? Simple: associate to the object c the set of all morphisms from c to other objects in C! Let's call this set hom(c,-).

Of course, this is more than a mere set: it's a set with extra structure. First of all, it's a set made of lots of little subsets for each object c' in C, we get a subset hom(c,c'), consisting of all morphisms from c to c'. Second of all, it's a set with an "action of C". In other words, given an element f in hom(c,c'), and a morphism g: c' -> c' ', we get an element fg in hom(c,c' '), just by composing f and g.

A set with all this structure has a name: it's called a "functor from C to Set".

The Yoneda lemma says that this "set with extra structure" knows everything you'd ever want to know about the object c.

Note that I got away with less than you might have thought I'd need! I only considered the morphisms *from* c, not the morphisms *to* c. In fact there is another version of the Yoneda lemma that uses the morphisms *to* c instead. I believe this is the one people usually talk about - but of course it doesn't really matter.

Great explanations can be found here:

Researcher

The motto in this section is: the higher the level of abstraction, the better.

Example1
Example2: