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Edit: Infos on the current state by Lieven Le Bruyn: http://www.neverendingbooks.org/grothendiecks-gribouillis

Edit: Just in case anyone still thinks that Grothendieck's unpublished manuscripts are (by his letter) entirely out of sight: Declared as "national treasure", they seem to be in principle accessible (+ Thanks to Jonathan Chiche who points - see his comment below - that it is not so clear if that idea was made a reality by now): http://www.liberation.fr/sciences/2012/07/01/le-tresor-oublie-du-genie-des-maths_830399

On p. 185 - 186 of the 3rd volume of Winfried Scharlau's Grothendieck biography, a handwritten text from 1986 by Grothendieck on foundations of topology, different from the concepts of topoi or tame topology, is shortly described. Scharlau doubts if it could be turned into a readable text, but perhaps someone knows the texts and has ideas about it?

Edit: Acc. to Winfried Scharlau's book, Grothendieck described his work in a letter to Jun-Ichi Yamashita as: "some altogether different foundations of 'topology', starting with the 'geometrical objects' or 'figures', rather than starting with a set of 'points' and some kind of notion of 'limit' or (equivalently) 'neighbourhoods'. Like the language of topoi (and unlike 'tame topology'), it is a kind of topology 'without points' - a direct approach to 'shape'. ... appropriate for dealing with finite spaces... the mathematics of infinity are just a way of approximating an understanding of finite agregates, whose structures seem too elusive or too hopelessly intricate for a more direct understanding (at least it has been until now)." Scharlau gives a copy of one page of the manuscript (at p. 188) and obviously has a copy of the complete text and remarks (on p. 199) that Grothendieck wrote a in 1983 letter about that theme to Z. Mebkhout.

Edit: In the meantime I could read a letter by Grothendieck about that, a summary: He started thinking from time to time about that ca. in the mid-1970's, the motivation was roughly that dissatisfaction with the usual topology which he expressed in the Esquisse, and looking at stratifications of moduli-"spaces" is his new starting point. Maybe, but not expressed in the letter or the Esquisse, the ubiquity of moduli problems in algebraic geometry (e.g. expressed in the beginning of Lafforgue's text ) is an other motivation. He describes his guiding ideas on new foundations of topology as more complicated than the guiding ideas behind the new foundations of algebraic geometry of EGA, SGA. A main test of his concepts now would be a "Dévissage"-theorem on "startified obstructions"(?) in terms of equivalences of categories. He has a precise heuristic formulation of that which helped him to find a "dévissage" corresponding to Teichmueller groups (probably what now is called "Grothendieck-Teichmueller group"?) which are related to stratifications "at infinity" of Deligne-Mumford moduli stacks.

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    $\begingroup$ I assume the answer is yes, and certainly was yes. Grothendieck! Please ask what you actually want to know. If this is impossible it is a strong sign this is not a good MO question. $\endgroup$
    – user9072
    Commented Nov 28, 2011 at 16:06
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    $\begingroup$ @quid: "real questions" is not an invariant notion, esp. if one takes the russian school concept of truth: "The main point is that the truth is a personality, not a mere object. Florensky formulates this in the framework of pure philosophy. For Florensky, an act of knowledge is a communication or relation, even a kind of "friendship" between the two persons, the one who studies and the one who is studied." as quoted by M. Harris in: people.math.jussieu.fr/~harris/MindBody.pdf Winfried Scharlau's remark on "readability" refers to a manuscript, the question is about thoughts. $\endgroup$
    – Thomas Riepe
    Commented Nov 29, 2011 at 16:30
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    $\begingroup$ If the text was written in 1986, that is about two years after "Pursuing Stacks", and the same year during which some parts of "Récoltes et Semailles" was written, several years before "Les Dérivateurs". Both "Pursuing Stacks" and "Les Dérivateurs" turned out to contain beautiful and important ideas, in spite of their having been neglected for years (at least for the first one) before some people realized it could be turned into something which everybody now find readable. (Even if these texts are not published yet, a part of the content is already available at least through expository texts.) $\endgroup$
    – Jonathan Chiche
    Commented Dec 2, 2011 at 16:46
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    $\begingroup$ (I once more insist that I do not find this a good question in its current form.) In the eighties, shortly after receiving "Pursuing Stacks", Bénabou ran a seminar in Paris in order to study the content of the text. I do not know what conclusion he drew himself, but I have heard several other participants of the workshop say that they just were not able to get through the difficulties. That nobody has been able to figure how to make a readable text out of the typeset version at the time did not mean nobody would ever be able to achieve that. $\endgroup$
    – Jonathan Chiche
    Commented Dec 2, 2011 at 16:53
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    $\begingroup$ I would assume that Scharlau believed Grothendieck, who pronounced all of his own manuscripts to be unreadable for anyone but himself. This has turned out not to be true, however. (In the meanwhile, Scharlau's biographies are available via Amazon btw, Parts I and III in German, Part I also in English.) $\endgroup$
    – Matthias Künzer
    Commented Mar 25, 2012 at 11:34

3 Answers 3

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In the light of past events ("Les Archives Grothendieck"), we now have:

Vers une Géométrie des Formes (1986)

  • I. Vers une géométrie des formes (topologiques) : notes manuscrites (05/06/1986).
    Cote n° 156-1 (26 p.)

  • II. Réalisations topologiques des réseaux : notes manuscrites (06/06/1986).
    Cote n° 156-2 (18 p.)

  • III. Réseaux via découpages : notes manuscrites (08/06/1986).
    Cote n° 156-3 (40 p.)

  • IV. Analysis situs (première mouture) : notes manuscrites (10/06/1986).
    Cote n° 156-4 (88 p.)

  • V. Algèbre des figures : notes manuscrites (14/06/1986).
    Cote n° 156-5 (48 p.)

  • VI. Analysis situs (deuxième mouture) : notes manuscrites (18-20/06/1986).
    Cote n° 156-6 (93 p.)

  • VII. Analysis situs (troisième mouture) : notes manuscrites (23-26/06/1986).
    Cote n° 156-7 (113 p.)

  • VIII. Analysis situs (quatrième mouture) : notes manuscrites (26/06-04/07/1986).
    Cote n° 156-8 (126 p.)

  • IX et IX bis. [Ateliers] : notes manuscrites (05-15/07/1986).
    Cote n° 156-9 (139 p.)

Project of transcription.

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  • $\begingroup$ Last link is broken. $\endgroup$
    – David Roberts ♦
    Commented Aug 1 at 3:38
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I cannot claim to have my own ideas about it, but Grothendieck's entire unpublished manuscripts; some 18,000 pages, were published last week by the University of Montpellier here:

I hope this is of help.

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    $\begingroup$ Not his "entire unpublished manuscripts" but only a small part of the Nachlass. $\endgroup$
    – Chandan Singh Dalawat
    Commented May 16, 2017 at 3:25
  • $\begingroup$ @ChandanSinghDalawat are you sure? I understood otherwise. There is a huge wealth of material. $\endgroup$
    – Robert Frost
    Commented May 16, 2017 at 6:18
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    $\begingroup$ Has anyone located the manuscript on tame topology? $\endgroup$
    – Arrow
    Commented Nov 4, 2017 at 23:53
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There is an article where you can find some ideas about this.

  1. Towards a new geometry of forms

In: The notion of space in Grothendieck: from schemes to a geometry of forms, John Alexander Cruz Morales arXiv

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