Finally, a math post! Prerequisites: basic category theory.

Sometimes, to understand an object in math, we consider ways the object can act on something else (e.g. a set), or ways the object can map into something else (or something else can map into it). For example, in group theory, it’s helpful to take a step back from looking at “a group ” to “all possible actions of a group on sets .” If we let the set be , i.e. we look at how acts on itself, we recover all information about .

Yoneda’s Lemma is a vast category-theory generalization of this idea. Instead of studying a category , we study functors from to (Sets). The functors from to (Sets) form a category–the functor category (the objects are functors and the morphisms are natural transformations). We can embed into this category , just as we can embed a group into the category of -sets by considering as acting on itself.

We state the lemma. (If you need help with the terminology, scroll down.)

**Yoneda’s Lemma:** Let be a locally small category. Let denote the functor and denote the contravariant functor (i.e. it is a functor ).

- (Covariant version) Let be functor from to (Sets). As functors , we have . ( is in , is in , and is a set.)
- (Contravariant version) Let be a contravariant functor from to (Sets). As functors , we have .

**Corollary (Yoneda Embedding):**

- The embedding given by sending is
*fully faithful.*(The morphism gets sent to $f\circ \bullet$.) - The embedding given by sending is
*fully faithful.*(The morphism gets sent to $\bullet\circ f$.)

**Remarks:**

- A category is locally small if homomorphisms between any two objects form a set.
- is the category of contravariant functors .
- has just the structure of a set.
- denotes the set of natural transformations between and .
- A functor is
*fully faithful*if is bijective for any objects and . This basically means that embeds the first category into the second, and there aren’t any “extra” maps between embedded objects that are present in but not . - We say a functor is
*representable*if for some (and ditto for contravariant).

**English?**

The statement basically tells us that the maps (natural transformations) between (i.e. the embedding of in the functor category), and arbitrary in the functor category is just given by , and ditto for the contravariant case. The corollary is more intuitive.

**Proof of Corollary:**

We show (2) of the lemma implies (2) of the corollary; (1) is entirely analogous. Set to get

Now a natural transformation is just a morphism in the functor category, so , and by definition , so we get

This is exactly the condition to be fully faithful.

**Example (Groups and group actions):**

Let’s return to groups acting on sets. Consider a group as a category with a single object, and all the group elements as morphisms from that object to itself, that compose in the way given by the group relations. Now what is an element of ? A functor sends an object to an object, so sends the single object in the group, say , to a set, say . It sends a morphism to a morphism (compatibly), so it sends an element of the group, say , in a contravariant way to a transformation , which has to be a bijection since is invertible. In a group action, each element of the group corresponds to a transformation (permutation) of the set , so *an element of is just a right group action on a set * (note contravariant-ness here gives the right action, though actually it depends on conventions). The Yoneda embedding thus embeds the single group into the category of all -sets, by sending to itself considered as a -set [*]. The fully faithful condition is

which recovers the following familiar fact: considering as a -set, the only -invariant homomorphisms from to are given by (multiplication by) elements of .

This example is often cited as “Yoneda’s Lemma is a generalization of Cayley’s Theorem” because by considering as a -set, we realize as a permutation group.

[*] More precisely, , the only object in which by abuse of terminology we consider as , is sent to . Since has only 1 object anyway, we identify this with just .

**Looking ahead**

If you haven’t seen it before, try proving Yoneda’s lemma. (It’s not too difficult if you’re familiar with basic category theory arguments… there is really only 1 choice for the map.) As a follow-up I recommend looking at Qiaochu Yuan’s article which gives additional examples: http://qchu.wordpress.com/2012/04/02/the-yoneda-lemma-i/

I’ll continue in a future post with the proof and how Yoneda’s lemma applies to algebraic geometry. Specifically, Yoneda Embedding allows us to jump between the viewpoint of a scheme with the viewpoint of “-valued points of ,” and knowing both viewpoints is especially useful in thinking about group schemes, as I learned in the first lecture of 18.787. (Will put up the lecture notes once I clear up some remaining confusions.)

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