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    • CommentRowNumber1.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 11th 2020

    Corrected the spelling.

    In English, “étale” in “étale space” is definitely an adjective, not a past participle (otherwise it would be “étaled space”).

    The last “e” is always silent in the English pronunciation, so cannot be an “é”.

    This also matches the usage in the literature.

    diff, v35, current

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 11th 2020
    • (edited May 11th 2020)

    I’m afraid you’re wrong here to “correct” the spelling.

    There’s lots of commentary about the two spellings. Here are two: here and here.

    The last “e” is always silent in the English pronunciation, so cannot be an “é”.

    As a speaker of (mathematical) English, if I am obliged to refer to étalé space or espace étalé, I pronounce that “e”, and so do other speakers I know.

    • CommentRowNumber3.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 11th 2020
    • (edited May 11th 2020)

    As a speaker of (mathematical) English, if I am obliged to refer to étalé space or espace étalé, I pronounce that “e”, and so do other speakers I know.

    espace étalé is French, so is not relevant in the discussion of English pronunciation.

    The standard reference on the subject, Mac Lane and Moerdijk, uses “étale space”, not “étalé space”.

    Johnstone explicitly states that he will not use this terminology at all.

    I could not find anything relevant in Kashiwara–Schapira, Bredon, Tennison, or Swan.

    What standard reference (preferably a book) uses “étalé space”?

    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 11th 2020
    • (edited May 11th 2020)

    It is relevant in terms of what native English speaking mathematicians actually say when they pronounce the word.

    Here are some course notes by Jonathan Wise. Other references I find might not be by category theorists, but algebraic geometers, many of whom draw a distinction in the spelling. (Who’s to say that Mac Lane and Moerdijk is the standard in such a discussion?)

    Edit: here’s Milne (see page 56 of 202).

    Grauert, Peternell, Remmert

    Rezk.

  1. FWIW, here’s a talk by Colin McLarty in which he speaks, from 4:43 to about a minute later, a bit about the étale/étalé terminology, including how Grothendieck pronounced these two words in his 1973 topos lectures.

    • CommentRowNumber6.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 11th 2020
    • (edited May 11th 2020)

    Re #4: Jonathan Wise says “etale space” (no accents) or espace étalé. I could not find “étalé space” in his notes, could you provide a precise reference?

    The same applies to Milne, who uses “espace étalé”.

    Grauert, Peternell, Remmert also use “espace étalé”.

    Rezk also uses “espace étalé”.

    So in other words, you are unable to cite a single instance of “étalé space” in the literature.

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 11th 2020
    • (edited May 11th 2020)

    Théo – okay, but that doesn’t eliminate the fact that many, many speakers of mathematical English use the more Francophone pronunciation.

    Please note that what I find wrong is Dmitri’s insistence that one way is wrong, against a lot of evidence of how many speakers speak when using mathematical English with loanwords. The original version before the edit was just fine.

    • CommentRowNumber8.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 11th 2020
    • (edited May 11th 2020)

    Also, the spelling étalé on Wikipedia was added there by Toby Bartels (I checked the editing history), who also added it here on the nLab. Perhaps Toby could tell us what his sources are.

    Dmitri’s insistence that one way is wrong

    Not “wrong”, just altogether absent from the published literature. If we are to create a totally new term, we should clearly indicate that it is not found in the published literature.

    • CommentRowNumber9.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 11th 2020
    • (edited May 11th 2020)

    Allow me to summarize the objective facts discovered so far:

    1) both “étale space” and “espace étalé” are present in the published literature.

    2) we have not yet found any mention of “étalé space” in the published literature.

    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 11th 2020
    • (edited May 11th 2020)

    So in other words, you are unable to cite a single instance of “étalé space” in the literature.

    Oh, come off it. Here (page 165). I’m certain I can find more.

    Here’s a third.

    • CommentRowNumber11.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 11th 2020

    Mentioned “espace étalé” as an alias.

    diff, v36, current

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 11th 2020

    Reinserted the undeniable fact that étalé space can be found in the literature.

    diff, v37, current

    • CommentRowNumber13.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 11th 2020

    Re #10: I am pretty sure that David Carchedi picked this spelling here on the nLab! We are in the situation of this comic here: https://xkcd.com/978/.

    • CommentRowNumber14.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 11th 2020

    I can easily find other instances. Dmitri, the fact of the matter is that this spelling is attested in the literature.

    • CommentRowNumber15.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 11th 2020

    Re #14: So far you have not provided any references to a published source, other than David Carchedi’s paper, which copied the incorrect spelling from the nLab, most likely.

    • CommentRowNumber16.
    • CommentAuthorThéo de Oliveira S.
    • CommentTimeMay 11th 2020
    • (edited May 11th 2020)

    @Todd_Trimble: Sorry, I should have phrased my message better; personally, I prefer étalé space too. (I just thought that the historical perspective there is quite interesting)

    Here is a Google search that searches specfically for étalé space and removes any results including etale or étale: [link]. Of course, this search doesn’t grab all results using “étalé space”, since any reference with the word étale is excluded. For instance, I couldn’t find Wedhorn’s Sheaves, Manifolds, and Cohomology there (which also uses étalé space).

    • CommentRowNumber17.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 11th 2020
    • (edited May 11th 2020)

    The parenthetical descriptive note is harmless and is correct as stated. I suggest that the matter now be dropped (except to note, hat tip to Théo, that Raymond Wells has this in his book – betcha anything he didn’t get it from the nLab).

    • CommentRowNumber18.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 13th 2020

    Re #17: Citing Raymond Wells’s book at the beginning would’ve made this discussion much shorter. I do have to say that Wells is by no means an expert on toposes and sheaves, so Mac Lane and Moerdijk’s terminology is far more authoritative to me.

    Speaking of the subject matter, is there a reference for étale spaces of sheaves of sets on arbitrary sites? Specifically, if F: S^op→Set is a sheaf of sets on a site S, then the étale space of F should be a geometric morphism of toposes Et(F)→Sh(S), where Sh(S) denotes the topos of sheaves of sets on S.

  2. I’d also be very interested in a reference for étalé spaces of sheaves on arbitrary sites. The only time I’ve seen this discussed was in MO78681 and here now…

    • CommentRowNumber20.
    • CommentAuthorTodd_Trimble
    • CommentTimeMay 13th 2020

    Dmitri, sorry, but I didn’t know about the Raymond Wells book at the beginning. I have to say that I was a little surprised that my assurance that some speakers (for example, geometers, as I mentioned in #4) do use the phrase was met with such strong skepticism, and beginning with #6, aggression as well. But I think we can put that discussion behind us now, and I’m glad to see the discussion shift to more mathematical matters.

    • CommentRowNumber21.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 13th 2020

    Re #20: Aggression is in the eye of the beholder, I guess, but allow me to point out that the thread started with my being accused of being wrong about changing to the terminology that was by far the most commonly used spelling in the published literature, with the only counterexample being an obscure book by Wells (obscure from the viewpoint of sheaves and toposes, not complex analysis). Considering that the only spellings that I have seen in numerous sources are “espace étalé” and “étale space”, it was only natural to inquire about the published source for “étalé space”—to which you responded with four different links, none of which had any mention of “étalé space” whatsoever.

    Re #19: It can certainly be the case that this general construction of étale space simply does not exist in the literature. (I asked David on MathOverflow.) However, I think we can simply transplant his answer to the nLab.

    • CommentRowNumber22.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 13th 2020

    Transplanted the MathOverflow answer.

    diff, v38, current

    • CommentRowNumber23.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 13th 2020

    Another bug in the nLab parser: the formatting is completely broken, even though no errors were indicated upon submission. Should I report this?

  3. @Dmitri: I think David might end up not seeing your comment; it seems the last time he logged on MO was on “May 28 ’19 at 12:29”…

    By the way, he has written a paper on étalé spaces of stacks: [arXiv:1011.6070], [published version], [errata] (note that the errata was published in 2016, after the last arXiv version).

    (I’m guessing this might be relevant to the discussion here; sorry if it isn’t!)

    • CommentRowNumber25.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2020
    • (edited May 14th 2020)

    I have fixed the rendering of the quote

    (by turning the double dollar-signs of the sub-environment into single dollar-signs)

    diff, v39, current

    • CommentRowNumber26.
    • CommentAuthorDmitri Pavlov
    • CommentTimeMay 26th 2020

    Added a reference.

    diff, v40, current

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