Category theory is not abstract nonsense

"General abstract nonsense" was coined by the people closest to the subject, and they said it with a grin. Category theory is what happens when you stop asking what mathematical objects are made of and ask only how they relate — and it turns out that is enough.

The phrase is usually credited to Norman Steenrod, one of the founders of algebraic topology, and it was a compliment: "nonsense" meant the parts of an argument that were formal, inevitable, and blessedly free of hard computation.¹ The machinery itself arrived in 1945, when @Eilenberg and @MacLane needed to make the word "natural" — as in "this isomorphism is natural" — into actual mathematics. Defining natural transformations required functors; defining functors required categories. The foundation was invented last, to hold up a ceiling that already existed.

The real content of the subject is a habit of definition. Ordinary mathematics defines objects by construction: a product is a set of ordered pairs, a quotient is a set of equivalence classes. The categorical habit is to define objects by #universal-property — by the role they play with respect to every other object at once — and to treat any construction as mere evidence that something fills the role.

Take the product. The categorical definition of A × B never looks inside it: A × B is whatever object comes equipped with two maps π₁ and π₂ back to A and B, such that any object X with its own pair of maps f to A and g to B admits exactly one map u into A × B consistent with everything in sight. That is the entire definition. It mentions no elements, no pairs, no coordinates — only arrows and one crucial word, unique.

The dashed arrow is the payoff. Because the definition mentions only arrows, it makes sense in any category, and it silently picks out the right object every time: cartesian products of sets, direct products of groups, product topologies on spaces — and, in the category whose objects are propositions and whose arrows are implications, the logical "and".² One definition, correctly instantiated across every field, with a bonus rigidly attached: any two objects satisfying it are isomorphic in exactly one compatible way. Constructions vary; the role does not.

Yoneda, honestly

The theorem that licenses all of this is the #yoneda-lemma, and it can be said in one honest paragraph. Fix an object A, and record, for every object X, the set of arrows from X to A — the totality of A's relationships, together with how those relationships transform when X varies. The lemma says this record determines A up to isomorphism, and that every consistent way of probing objects by arrows is itself given by an object. Nothing about A is lost by discarding its internals and keeping its interactions. Definition by relationship is not a stylistic preference; it is mathematically complete.

That is why "abstract nonsense" settled into affection rather than insult. Grothendieck rebuilt algebraic geometry on exactly this #naturality-first foundation, and the same diagrams now type-check programs and organise databases.³ The nonsense is the load-bearing part: what survives when every specific detail is stripped away is precisely the content that transports between fields intact.