Andrej Bauer has a paper titled The pullback lemma in gory detail that goes over the proof of the pullback lemma in full detail. This is a basic result of category theory and most introductions leave it as an exercise. It is a good exercise, and you should prove it yourself before reading this article or Andrej Bauer’s.
Andrej Bauer’s proof is what most introductions are expecting you to produce. I very much like the representability perspective on category theory and like to see what proofs look like using this perspective.
So this is a proof of the pullback lemma from the perspective of representability.
The key thing we need here is a characterization of pullbacks in terms of representability. To
just jump to the end, we have for
That is to say we have the natural isomorphism
We’ll write the left to right direction of the isomorphism as
One nice thing about representability is that it reduces categorical reasoning to set-theoretic reasoning that you are probably already used to, as we’ll see. You can connect this definition to a typical universal property based definition used in Andrej Bauer’s article. Here we’re taking it as the definition of the pullback.
The claim to be proven is if the right square in the below diagram is a pullback square, then the left
square is a pullback square if and only if the whole rectangle is a pullback square.
Rewriting the diagram as equations, we have:
Theorem: If
Proof: If
The second isomorphism is
This overall natural isomorphism, however, is exactly what it means for
We simply run the isomorphism backwards to get the other direction of the if and only if.
The simplicity and compactness of this proof demonstrates why I like representability.