Grothendieck famously objected to the term "perverse sheaf" in Récoltes et Semailles, writing "What an idea to give such a name to a mathematical thing! Or to any other thing or living being, except in sternness towards a person—for it is evident that of all the ‘things’ in the universe, we humans are the only ones to whom this term could ever apply.” (Link here, in an excellent article by Allyn Jackon.) But a google search for '"perverse sheaf" etymology' gives only nine hits, none of which seem informative.

What is the etymology of the term "perverse sheaf"?

  • Wikipedia explains the "sheaf" part and says that the "perverse" part comes from intersection homology: – Qiaochu Yuan Jun 29 '10 at 21:19
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    Well the sheaf part I get; saying this comes from intersection homology seems to beg the question. – Daniel Litt Jun 29 '10 at 21:22
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    I just meant that the mysterious part of the question could be rephrased "what is the etymology of the term 'perversity'?" – Qiaochu Yuan Jun 29 '10 at 22:23
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    BTW, I believe I have read that Goresky and MacPherson wanted to define "perverse homology", but Deligne objected, and so we get the unfortunate term "intersection homology". Whenever I talk to non-topologists about multiplying cohomology classes on a manifold by intersecting their Poincar\'e duals, I get asked "is that what `intersection homology' refers to?" and curse G&M for not sticking to their guns. – Allen Knutson Jun 30 '10 at 1:22
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    @AK - google says Sullivan, not Deligne. (I asked google b/c I didn't believe D, since BBD promoted the word!) – Ben Wieland Jun 30 '10 at 2:47
up vote 111 down vote accepted

When MacPherson and I first started thinking about intersection homology, we realized that there was a number that measured the "badness" of a cycle with respect to a stratum. This number had the property that when you (transversally) intersected two cycles, their "badness" would add. The best situation occurs for cocycles, in which case that number was zero, and the intersection of two cocycles was again a cocycle. The worst situation was for ordinary homology, in which case that number could be as large as the codimension of the stratum. In that case, the intersection of two cycles could even fail to be a cycle. After a while it became clear that we needed a name for this number and we tried "degeneracy", "gap", etc., but nothing seemed to fit. It seemed that the bad cycles were being "obstinate", but "obstinateness" did not sound reasonable. Finally we said, "let's just call it the perversity, and we'll find a better word later". We tried again later, with no success. (We did not realize that in some languages the word is obscene.) When we first went to talk with Dennis Sullivan and John Morgan about these ideas, we were calling the resulting groups "perverse homology", but Sullivan suggested the alternative, "intersection homology", which seemed fine with us. This was 1974-75. Later, when it was discovered that, for any perversity, there is an abelian category of sheaves, whose simple objects are the intersection cohomology sheaves (with that perversity) of closures of strata, Deligne coined the term "faisceaux pervers".

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    While thankfully unaware of the details, I suspect that Mark Goresky means "obscene" when he says "profane." – Michael Thaddeus May 16 '14 at 4:21
  • Changed, thanks! – mark goresky Jul 19 '14 at 1:56
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    What a cheap way to get reputation points: just invent a brilliant chunk of mathematics and 40 years later reap the benefits by remembering why you chose the terminology... – Georges Elencwajg Jul 19 '14 at 9:35
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    Tate's unique Mathoverflow answer is of the same nature.… – Noam D. Elkies Jun 4 '15 at 2:33

One explanation I heard (it may have been from MacPherson but I am not sure) was that "perverse" was used in the sense of "contrary", the cycles used in the definition refuse to move away from the singularities.

"Goresky and MacPherson relaxed the transversality condition on the cycles by allowing them to deviate from dimensional transversality to each stratum of codimension k, for each k > 2 (by hypothesis there are no strata of codimension 1), within a tolerance specified by a function p(k), which they called the perversity." From the book review of Kirwan's book on intersection homology.

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    Why is it called perversity? – Daniel Litt Jun 29 '10 at 21:24
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    The review you reference does have the following wonderful quote however: "Because of all those marvelous properties, everyone calls these special complexes...*perverse sheaves*. Of course, they are complexes in a derived category, not sheaves. And, they are well behaved, not perverse. Nevertheless, the name has stuck." – Daniel Litt Jun 29 '10 at 21:30
  • Similarly, in the May 2010 AMS Notices (free online at the Webpage, de Cataldo and Migliori answer the question What is a .... perverse sheaf? but resort to roughly the same circular reason to explain the terminology. (This note, though longer than the usual couple of pages for that column, was no doubt solicited after their very long and useful survey appeared recently in the AMS Bulletin.) Good luck to those who want a rational explanation. – Jim Humphreys Jun 29 '10 at 22:36
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    See page one and footnote of Goresky & MacPherson's notes here – Justin Curry Jun 29 '10 at 22:49
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    My adviser has been known to remark that there are two types of people in the world, those who like singular spaces and those who don't. If you're part of the latter category (my impression being that the majority of people, especially in the 70's, are), then the term perversity certainly makes sense. The perversity function essentially tells you how far intersection cochains are allowed to be from singular cochains, a notion you would probably only consider if you're working with singular spaces. – Mike Skirvin Jun 29 '10 at 23:54

When I was a grad student, a classmate commented about the odd term "perverse" in IH but I defended it. I thought I had figured out the rationale, which I denied had anything to do with perversity in the ordinary sense of that word. My theory was that the "verse" was from "transverse," which is a generic kind of intersection of objects of complementary dimension. By replacing "trans" with "per" we get a more general notion of intersection. The "per" here meant excess or abundance, following the way chemists use that prefix in words like peroxide or permanganate. A perversity in IH in fact specifies how far in excess of the generic expectation the dimension of an intersection with a stratum is allowed to be. My classmate was skeptical, but this theory seemed so plausible to me that I never really doubted it. I am genuinely surprised to be finally refuted.

Here's what I like to think, me and my ignorant self. Why name something after a word? Most folks would do it because they think that whatever meanings and connotations that word already carries would apply well to the concept being named. I see your quoted Grothendieck as a strict practitioner, and indeed a master, of this style of naming: think of "etale", "crystalline", "topos"...

However, this point of view ignores an important aspect of the naming process -- its bidirectionality. Once you name something after a word, the word is forevermore changed in its meanings, connotations, usage, and cultural presence, by simple virtue of being attached to this thing it wasn't attached to before. That is, it's possible to view the act of naming not as applying a word to an object, but applying an object to a word.

Coming to perverse sheaves, the question was why such "beautiful" and "well-behaved" objects deserved the name perverse. Certainly from Grothendieck's perspective on naming this seems to be a travesty; but from the second perspective it makes perfect sense: how better to soften the harsh and pejorative word "perverse", at least in certain (mathematical) circles, than to apply it to such fantastic objects?

I view the naming of perverse sheaves as a brilliant and subversive act.

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    Dustin: This is an interesting interpretation, but have you any evidence? ;-) – Kevin H. Lin Jun 30 '10 at 2:58

The word "perverse" has strong and jarring connotations in some languages, such as German, but became standard usage nonetheless.

One of the founders of the theory said that the term was unpopular with everybody except for one specific mathematician --- who was another of the famous founders.

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    The remark by one of the anonymous "founders" is strongly reminiscent of the amusing footnote to the 1965 Borel-Tits IHES paper in which one author apologizes for the use of the then-standard term *Borel subgroup", having capitulated only because the other (unidentified) author insists on it. Both Goresky and MacPherson are now at IAS as Borel once was, and knew him well along with his work – Jim Humphreys Jun 29 '10 at 22:50
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    We, mathematicians, have become oblivious of jarring connotations. I recall this whenever I mention pathological (anything) in class and meet funny looks of the students. – Victor Protsak Jun 30 '10 at 8:46
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    T., while multiple people contributed to the origins of perverse sheaves as a useful concept, the word "perverse" certainly comes from intersection homology, which was officially developed by Goresky and MacPherson (though certainly with some contributions from Sullivan, McCrory, and others...) – Greg Friedman Jul 4 '10 at 7:49
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    The originator's comment referred specifically to the term "perverse sheaf" (and did not imply any role in creating the term). I don't think that the use of perversities in intersection homology raises as many linguistic hackles, because (1) it doesn't involve constantly speaking of any specific objects as being perverse, and (2) it can be softened by speaking of a "perversity function" or "perversity data", not a perversity as such, and (3) "middle perversity" is more clearly humorous than perversity itself. – T.. Jul 5 '10 at 17:18

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