21. Natural transformations and the Eilenberg-Steenrod Axioms

In this lecture we finally give a precise meaning to the word "naturality".

Pro Tip: You can now annoy all your other lecturers by sticking your hand up in class up as soon as they say something like "and so blah-blah $X$ is naturally isomorphic to $Y$", and demanding to know the precise categorical setting. "Excuse me, Professor, could you please tell me exactly which natural transformation you had in mind?"^[1]

We then stated the famous Eilenberg-Steenrod Axioms for a homology theory, and formulated the main "existence and uniqueness" result. This is too hard to prove, so we finished the lecture by stating a "baby" version (albeit still highly non-trivial). We will prove the baby version on Friday.