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 $\bullet\circ f$.)
- 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).
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 .
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.)