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I have created the entry recollement. Adjointness, cohesiveness etc. lovers should be interested.
notice that we have a (stubby) entry six operations
I made a mistake. It is not directly related sorry. Six operations are an issue about Grothendieck duality, not recollement.
When Grothendieck talks about the yoga of six operations than he means a heuristic meta-theory encompasing a variety of contexts, not the single axiomatic framework – it is a general story about the derived duality in mathematics. The recollement seems introduced by Beilinson-Bernstein-Deligne much later and is a much simpler issue.
Why is the word “recollement” not translated into English? E.g. google translate suggests “reattachment” or “gluing”.
I would prefer that it does stay so. Gluing is taken often as a synonym of a descent, while recollement is a more specific situation, and it is good that the term is referred to with a standard and technical term. Reattachement would be a better candidate as, unlike gluing, it is not used in math much so far, but so many English papers already use ”recollement”.
How do English speakers pronounce it?
What I hear is the same as in French, but without nasalization and with true n.
Not being a French speaker myself, that doesn’t help me. (-:
rəkɔlman vs. rəkɔlm
As a bilingual speaker (;-)), if I was saying it, it would correspond to a spelling of ’recolment’ i.e. more or less as in French but sounding the nt in ment.
I never heard the nt pronounciation so far in English language talks on the subject (the word comes pretty often in algebraic geometry talks). I just hear identical to French without nasalization. I do not know of the usages out of mathematics.
You will have to get me to say it to you next time we meet! Often English speakers adopt a pseudo-French pronunciation and that is probably what you have heard.
Right, so what is wrong with it ? It is still written French and sc ientists use the pseudo-French quite well, while in Schrift we can italicize those. Even on cuneiform tablets embedded foreign words used to be italicized in their graphics: Glossenkeil consisting of two Winkelhaken signs was used before a phrase quoted in another language. This witnessed that speakers of natural languages even in dawn of literacy were aware of embedding status and were comfortable with conventions of switching.
FWIW, the analogous topos-theoretic context is almost always called (Artin) gluing, and I haven’t ever heard anyone complain about a clash between that sort of gluing and sheafy gluing.
You say that I should learn what “FWIW” is, but I should not learn word recollement ?
Well, FWIW is an abbreviation of an English phrase.
Also, I would never use FWIW in a mathematical paper, much less as a mathematical definition.
But abbreaviations are not a part of natural (spoken) language, nor of standard orthography (apart from few like i.e.), they are the convention of limited communities which do not pretend to present the general language corpus. Look at wikipedia for D.M.V. for example, it denotes more than 10 different phrases. I know it as my former director needed to be called by D.M.V.
Strangely enough both communist SSSR and USA have big like for abbreviations, far more than German and especially far more than French, and I can say, certainly far more than Croatian. Dieudonne was once writing something about this while defending the terminology of Bourbaki. Russia now has shorthands for English phrases like terakt for terrorist act and so on; terrible damage to their language.
Mathematical expressions and notation tend to borrow from many languages and scripts. Logicians like Hebrew notation, to mention the most notable example. Russian took for example сайт (sajt, or if you prefer, sayt) for (Grothendieck) site, which is transferred hence via a phonetic/pronounced version, and it is now entirely standard. I am now suggesting it into Croatian math terminology, the same way, sajt.
I’m not sure what your point is in #18.
I think it’s fine for mathematics to use different alphabets for symbols (there aren’t enough letters in one alphabet), but I don’t think it’s a good idea to use words from another language; they should be translated into the language in which you are writing. But “recollement” seems pretty well established even in English writing about these things, so probably it is not going to get changed.
18 was just a comment illustrating that while you are absolutely right that FWIW may be understood as an abbreviation of an English phrase, it has likely not achieved a status of an abbreviation which is a part of the standard (general) English language sign, and that alleviating too many abbreviations to that status if it is going to happen would be of a debatable value.
I have cross-linked recollement and adjoint modality, and have added an Example-item at “recollement”: global equivariant stable homotopy theory.
(coming back to this four year old thread here: #15 is pretty funny)
The recollement page uses (resp. ) for the open (resp. closed) embedding, while Artin gluing uses the reverse. Is there a good way to reconcile them? I can make the edits if there’s agreement.
I think that there are some simple typos in the recollement exact triangles for the triangulated case. Namely, I think should be and should be . In the classical context of sheaves on a topological space with an open embedding and the complementary closed embedding, the triangles would make sense and be correct, I think, but it doesn’t make sense in the abstract setting.
Chris Brav
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