Editing Abstract nonsense (section)

==Background==

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 proof]]s that rely on category-theoretic ideas often seem out-of-context, somewhat akin to a [[non sequitur (literary device)|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,<ref name="monastyrsky">Michael Monastyrsky, ''Some Trends in Modern Mathematics and the Fields Medal.'' Can. Math. Soc. Notes, March and April 2001, Volume 33, nos. 2 and 3.  Online version available at http://www.fields.utoronto.ca/aboutus/FieldsMedal_Monastyrsky.pdf.
:"''In algebra, the term “abstract nonsense” has a definite meaning without any pejorative connotation.''"</ref><ref name="mathworld" /> and is instead used jokingly,<ref name="maclane"/> in a [[self-deprecating]] way,<ref name="rotman"/> affectionately,<ref name="lang"/> or even as a compliment to the generality of the argument. [[Alexander Grothendieck]] was critical of this notion, and stated that:
{{Citation bloc|The introduction of the [[zero|cipher 0]] or the [[Group (mathematics)|group]] concept was general nonsense too, and mathematics was more or less stagnating for thousands of years because nobody was around to take such childish steps... <ref>{{cite web |title=Correspondance Alexander Grothendieck - Ronald Brown |url=https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircle/Letters/LettersGrothendieckRBrown.pdf |publisher=Société Mathématique de France}}</ref>}}

Certain ideas and constructions in mathematics share a uniformity throughout many domains, unified by category theory. Typical methods include the use of [[classifying space]]s and [[universal property|universal properties]], use of the [[Yoneda lemma]], [[natural transformation]]s between [[functor]]s, and [[diagram chasing]].<ref>{{Citation|last=Marquis|first=Jean-Pierre|title=Category Theory|date=2019|url=https://plato.stanford.edu/archives/fall2019/entries/category-theory/|encyclopedia=The Stanford Encyclopedia of Philosophy|editor-last=Zalta|editor-first=Edward N.|edition=Fall 2019|publisher=Metaphysics Research Lab, Stanford University|access-date=2019-10-27}}</ref>

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.<ref name="mathworld">{{MathWorld| urlname=AbstractNonsense | title=Abstract Nonsense|author=Macura, Wiktor K.}}</ref> For example, one might say that "By abstract nonsense, [[product (category theory)|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 one to skip proof details that can be considered trivial or not providing much insight, focusing instead on genuinely innovative parts of a larger proof.