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    • CommentRowNumber1.
    • CommentAuthorThomas Holder
    • CommentTimeMar 25th 2014
    On the nlab I see occasionally (referee on Tom's paper, entry on 'localic topos') reference to Chicago lecture notes of Lawvere as 'Variable sets, etendu, and variable structures in topoi' from 1976. The copy available to me reads 'Variable sets etendu and variable structures in topoi', dated 1975. As neither title appears on Lawvere's publication list I wonder whether there really IS a 1976 version with altered title around ?

    Concerning the spelling 'etendu/e':
    my understanding is that the concepts originates as the French feminin noun 'l'étendue' in SGA4 where the meaning alludes to 'extension, generalization, breadth, flatness' indicating thereby a tame variation on the concept of sheaf topos on a topological space. In the title of Lawvere's notes it is used as an adjective 'sets etendu' as backtranslation from hypothetical French 'ensembles étendus' (actually, it seems, that in Grothendieck only the feminin noun from occurs). In the notes themselves Lawvere uses exclusively the noun form 'etendue' as do all other researchers (Rosenthal, Kock&Moerdijk, Lawson&Steinberg) working on the concept I am aware of. In a comment for the TAC reprints Lawvere uses 'etendu' as a noun whereas the article he comments on itself uses 'etendue'. Nevertheless, I figure that the preferred use in English should be 'etendue' (a neuter noun!?) whereas the adjectival form should be avoided altogether. Maybe some expert in English or topos theory could dis/conform this !?

    Concerning the concept:
    my guess is, that Lawvere's attention to the concept has been drawn by Grothendieck during the latter's 1973 visit in Buffalo (In the Cardone(?) interview Lawvere describes the content of their discussions as instructions on the 'points have symmetries'-principle in algebraic geometry). So the concept shows up in Lawvere 1975 lecture note and the article for the Eilenberg-Festschrift where he accordingly stresses the 'spinning points' available in an etendue. In the 1980s Lawvere emphasizes the concept as paradigmatic 'petit topos' of processes where the then available general (all monic) site description is interpreted as precondition for process identity.
    I have the impression that meanwhile the topos community has lost interest in the concept because of some results in the 1980s showing that more or less every topos is just a covering away from being sheaves on a space. Or is it, that the site characterization says everything there is to say !? So my question would be whether somebody could expand or comment on this and/or point to literature beyond the above mentioned authors (Lawvere mentions somewhere in the 1980/90s that Grothendieck's terminology changed 'étendue' to 'topological multiplicity' by then, that must presumably be in the manuscripts surrounding Pursuing Stacks !?) What has become of the petit/gros/etendue concepts in the French school of algebraic geometry -are they still in use ?
    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 25th 2014

    Thomas, re the first paragraph: obviously it was a mistake (mine); I was going on memory without looking it up. I’ll fix it. You should always feel free to correct obvious errors! :-)

    I didn’t know it was being used as an adjective in the title; I knew there was a noun like that in the text, but I was unaware it was feminine. Good to know.

    • CommentRowNumber3.
    • CommentAuthorThomas Holder
    • CommentTimeMar 25th 2014
    It certainly didn't look like an 'obvious' mistake to me. Me being neither French nor English, Lawvere's use in 'variable sets etendu' is rather intransparent: it looks like a postponed adjective respecting the French order but disregarding the French agreement requirement, personally I would render it as 'variable sets étendus' but then I am neither French nor English. The way you interpreted it, I would use the plural 'variable sets, etendues, ..) but then I am not English!
    • CommentRowNumber4.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 25th 2014

    (The obviously wrong stuff in the nLab was the date, the commas, and the spelling of the noun.)

    Please have a look at localic topos to see whether I transcribed your (very useful!) comment correctly – there’s a footnote that explains. I also fixed the review for Leinster’s article.

    I am not aware of any work on etendues post those Lawvere notes.

    • CommentRowNumber5.
    • CommentAuthorTim_Porter
    • CommentTimeMar 25th 2014
    • (edited Mar 25th 2014)

    Todd: try Pedro’s paper and, by Anders and Ieke, Presentations of étendues.

    • CommentRowNumber6.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 25th 2014

    Okay, thanks Tim.

    • CommentRowNumber7.
    • CommentAuthorThomas Holder
    • CommentTimeMar 25th 2014
    I guess that this reflects my ideas. I'd rather not link the expression to the 1973 conversations though because this is just a speculative conceptual link and doesn't seem to be related the expression in the title of the lecture notes. It simply occurs to me that the concept hasn't played a role in Lawvere's writing before 1973, and the spinning points of the etendues figure prominently in 1975. In the 1980s there is a shift and etendues are viewed by Lawvere as one petit class together with the class that has an all epic site representation (I guess in the Como proceedings and the 'qualitative distinctions' paper): he stresses the cancellation properties in the sites and links this to the agreement of different observers on process identity. This seems to be tied to the reconfiguration of the landscape along the petit/gros axis.

    Concerning the 'etendu' let me stress that as non native speaker it is rather obscure to me: I don't have a good intuition what is the correct way to translates French terms into English, what happens to syntax, gender, etc. As far SGA4 is concerned it appears there on page 482 (in one of the famous 'exercises') in nominal form as 'une étendue (topologique)'. So my guess is that it should it be translated as a '(topological) extension' or in 'Frenlish' as a '(topological) etendue', that also seems to be the consensus in the English literature (including Lawvere himself in most of the cases). The masculin form in the title of Lawvere's lecture notes might be only a mistake by the typist (I once saw some graph theoretical results of Lawvere attributed to a 'Lover' which I presume is not really a graph theorist unknown to me!). How easy it is to make mistakes, I had to realize when I saw that in the very post where I was interrogating on the correct form of the Lawvere paper above I actually switched 'variable structure' to 'variable structures'!
    • CommentRowNumber8.
    • CommentAuthorThomas Holder
    • CommentTimeMar 25th 2014
    I wasn't aware of the Resende reference, thanks from here too! The latest reference I was aware of is Lawson-Steinberg 'ordered groupoids and étendues' in 2004 45(no.2) cahiers (sorry, I have troubles with copying the link!). The elephant (II,769ff) has his saying, too.
    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeMar 25th 2014

    Okay, the footnote has been pared down a little more.

    • CommentRowNumber10.
    • CommentAuthorTim_Porter
    • CommentTimeMar 25th 2014

    Thomas, Ask Pedro Resende for more details and any more recent stuff. When last I was in Lisbon he talked a little about this with me, but I forget the direction he was looking at. You could also ask Mark Lawson and Ben Steinberg.

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 25th 2014

    Note also the paper of Pronk in Composito in 1996 on bicategorical localisation deals with etendues (spelling?)

    • CommentRowNumber12.
    • CommentAuthorThomas Holder
    • CommentTimeMar 25th 2014
    Spelling is fine! Thanks for the reference!
    (If I've might given the impression of being deeply worried by spelling of concepts or names of great mathematicians, please believe me, I am not. I've brought up 'etendu/e' with the idea of eventually writing a seperate entry for it on the nlab although at this point I feel that this somewhat beyond my expertise as I lack even the ground for choosing among the possible definitions - and probably contrary to appearance even choosing a spelling!)

    I'll definitely have a closer look at the Resende paper too, although I've probably flipped through it when it was laid on the arxiv but have failed to pay attention to the appearance of the 'etendue'-concept then unfamiliar to me.
    • CommentRowNumber13.
    • CommentAuthorThomas Holder
    • CommentTimeMar 28th 2014

    I guess there’s still a while until I have the courage to write a seperate entry for ’etendue’ (or somebody else finds the energy) and as this discussion strang has to serve provisorily as a placeholder of at least the reference section of such a hypothetical entry, let me add the following references to complete the picture: In the aftermath of his phd. Kimmo Rosenthal has published a series of papers concerning etendues, among which

    Rosenthal, Étendues and categories with monic maps, J Pure Appl. Alg. 22 (1981)

    has a nice exposition of the development of the concept from Grothendieck to Lawvere.

    The most recent exposition of Lawvere’s views of etendues&les petits can be found in the 2008 Como lectures:

    Cohesive Toposes –Combinatorial and Infinitesimal Cases

    (A side remark on that entry: Lawvere uses the term ’category of cohesion’ in the TAC 2007 paper because there he considers cases like the Hurewicz homotopy category that are not toposes but as quality types are still ’extreme cases’ of cohesion. The possibility to accomodate non-toposes is crucial (possibly not only!) for the paper. So in my view the use of the term ’cohesive topos’ in Como is only a minor variation on the terminology of the TAC paper in a context where toposes are in focus.)

    • CommentRowNumber14.
    • CommentAuthorTim_Porter
    • CommentTimeMar 28th 2014
    • (edited Mar 28th 2014)

    Kimmo did his thesis on them: see here

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