Abstract nonsense

In mathematics, abstract nonsense, general abstract nonsense, and general nonsense are terms used by mathematicians to describe abstract methods related to category theory and homological algebra. More generally, “abstract nonsense” may refer to a proof that relies on category theoretic methods, or even to the study of category theory itself.

Roughly speaking, category theory is the study of the general form, that is, categories of mathematical theories, without regard to their content. As a result, mathematical proofs that rely on category theoretic ideas often seem out of context, somewhat akin to a non sequitur. Authors sometimes dub these proofs “abstract nonsense” as a light-hearted way of alerting readers to their abstract nature. Labeling an argument "abstract nonsense" is usually not intended to be derogatory, and is instead used jokingly, in a self-deprecating way, affectionately, or even as a compliment to the generality of the argument.

Certain ideas and constructions in mathematics share a uniformity throughout many domains, unified by category theory. Typical methods include the use of classifying spaces and universal properties, use of the Yoneda lemma, natural transformations between functors, and diagram chasing.

When an audience can be assumed to be familiar with the general form of such arguments, mathematicians will use the expression Such and such is true by abstract nonsense rather than provide an elaborate explanation of particulars. For example, one might say By abstract nonsense, products are unique up to isomorphism when they exist, instead of arguing about how these isomorphisms can be derived from the universal property that defines the product. This allows to skip proof details that can be considered trivial or not providing much insight, focusing instead on genuinely innovative parts of a larger proof.

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