Anon, a young researcher asks Daniel Kan about category theory.
Anon: Dr. Kan, can you please explain the Yoneda lemma from category theory to me in terms a normal person can understand?
Daniel Kan: Oh, sure. The Yoneda lemma is one of the most famous results in category theory and it's actually pretty simple. Basically what happens is that we have two categories C1 and C2 . In C1 , there are objects x 1 , …, x n such that for all i ∈ C1 (x 1 ′ = x 2 ′ +…+x n ′), if f(i) then g(i). For example, consider the set {0, 1, 2} which consists of numbers 0 through 9. Now suppose we want to find all sets with members equal to 5. So, the objects in C1 would be the set of numbers 0 through 9. In C2 there are objects A, B and C. In this case, C1 would be the set of natural numbers. So we want to find all subsets of the set of natural numbers with members equal to 5. So in C2 we have the objects A, B and C. In this case, f(x 1 ′) = x 2 ′ and g(x 1 ′) = x 3 ′. So, x 1 ′ would equal to the number 0 and x 2 ′ would equal to the number 5. This is true for all other members of the set of natural numbers. So if f(i) then g(i). So, all sets with members equal to 5 are either in the set of numbers 0 through 4 or the set of numbers 1 through 6. So objects in C2 are subsets of the set of natural numbers. Now, in the Yoneda lemma, we have that C1 is the category of all sets and C2 is the category of all topological spaces. It is all quite elementary, really.
Anon: So in English, what is the Yoneda lemma?
Daniel: The Yoneda lemma says that, if we have a category with a bunch of objects and morphisms and then there is another category with a bunch of objects and morphisms. Then, for every object in the first category there is a unique morphism in the second category such that the following diagram commutes.
He proceeds to scribble down a complicated diagram.
Anon: So, how do I apply this in my research?
Daniel: Well, you don't.