Folklore and triviality in category theory
Category theory is known to have quite a number of ‘folklore results’. In this context, a ‘folklore result’ is one that (supposedly) “everyone knows”, even though no-one has ever actually published a proof in The Literature, usually because it's ‘obvious’ or ‘trivial’.
The nLab, which was partly founded - or so i understand - in an effort to address this phenomenon, has an entry for ‘folklore’[a]. It quotes Paul Taylor describing ‘folklore’ as:
a technical term for a method of publication in category theory. It means that someone sketched it on the back of an envelope, mimeographed it (whatever that means) and showed it to three people in a seminar in Chicago in 1973, except that the only evidence that we have of these events is a comment that was overheard in another seminar at Columbia in 1976. Nevertheless, if some younger person is so presumptuous as to write out a proper proof and attempt to publish it, they will get shot down in flames.
The entry goes on to quote Clark Barwick discussing homotopy theory / algebraic topology:
The people at the top of our field do not, as a rule, issue problems or programs of conjectures that shape our subject for years to come. In fact, in many cases, they simply announce results with only an outline of a proof – and never generate a complete proof. Then, when others work to develop proofs, they are not said to have “solved a problem of So-and-So”, rather, they have “completed the write-up of So-and-So’s proof” or “given a proof of So-and-So’s theorem”. The ossification of a caste system – in which one group has the general ideas and vision while another toils to realize that vision – is no way for a subject to flourish.
It's said that in category theory, once one actually groks particular concepts and definitions, results can become ‘obvious’ or ‘trivial’. In my own (admittedly very limited) experience in learning category theory, this seems to be somewhat true[b].
The problem is that the field of category theory is, to use Richard Feynman's phrase, full of “monster minds”[c]. Results which seem ‘obvious’ or ‘trivial’ to such people can seem much less so to others, who then have to expend extra effort on trying to understand them. On the one hand, it's understandable that Monster Minds won't want to spend time on writing out - or having a grad student write out - the details of a proof they feel is ‘obvious’ or ‘trivial’. On the other hand, without such a proof, it can be difficult for us lesser beings to gain insight and understanding into why a result is true (which is basically the point of a proof, or at least a non-formal one).
It can also be argued that a ‘folklore’ result isn't actually a result - merely a ‘conjecture’ - until a more-than-handwavy proof has been published, since it might turn out that things aren't as straightforward as they seem. The Jordan curve theorem, which roughly states that a simple closed curve on the plane (such as a circle) divides the plane into an ‘inside’ and an ‘outside’, seems obvious, yet is non-trivial to prove[d]. And famously, a paper by Fields medallist Vladimir Voevodsky, “Cohomological Theory of Presheaves with Transfers”, contained a mistake in the proof of a key lemma, yet was studied and used by multiple mathematicians before the mistake was noticed[e]. As Voevodsky wrote:
A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail.
It seems to me that publishing a rigorous proof of a folklore result is something that should be welcomed, as long as the author(s) is/are not claiming priority. Knowing whether or not something is true can be less significant in the long run than knowing why it is or isn't true, as the techniques used can potentially be leveraged to solve other problems. Clear expositions of why a folklore result is true can make it easier for people to learn and contribute to mathematics.☙
[b] For example, the amusing moment when i realised that a monad is just a monoid in the category of endofunctors. :-) Background:
[c] Cf. the chapter “Monster Minds” in “Surely you're joking, Mr Feynman!”, in which Feynman relates how the first seminar he presented was attended by not only Wheeler, for whom he was a research assistant, but also Wigner, von Neumann, Pauli and Einstein.