Why is it interesting?

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

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

• http://www.math3ma.com/mathema/2016/9/12/the-most-obvious-secret-in-mathematics
• http://www.math3ma.com/mathema/2017/8/30/the-yoneda-perspective

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

• http://www.math3ma.com/mathema/2017/9/6/the-yoneda-embedding
• 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.

John Baez

Researcher

Examples

Example1

Example2:

FAQ

History