The Pullback Lemma in Gory Detail (Redux)

Introduction

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.

Preliminaries

The key thing we need here is a characterization of pullbacks in terms of representability. To just jump to the end, we have for and , is the pullback of and if and only if it represents the functor

That is to say we have the natural isomorphism

We’ll write the left to right direction of the isomorphism as where and and they satisfy . Applying the isomorphism right to left on the identity arrow gives us two arrows and satisfying and . (Exercise: Show that this follows from being a natural isomorphism.)

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.

Proof

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 , , and is a pullback of and , then is a pullback of and if and only if is a pullback of and .

Proof: If was a pullback of and then we’d have the following.

The second isomorphism is being a pullback and is an arrow into so it’s necessarily of the form . The first equality is just mentioned earlier. The second equality merely eliminates the use of using the equation .

This overall natural isomorphism, however, is exactly what it means for to be a pullback of and . We verify the projections are what we expect by pushing through the isomorphism. By assumption, and will be and respectively in the first isomorphism. We see that .

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.