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I have a question about the the intuition of what Grothendieck proposed as tame topology in his "Esquisse d’un programme" as a "better suited" geometric structure in order to have a better control over spaces using stratification techniques.

My question is about which "picture" one should have in mind on the interpretation that this hypothetical topology should have a "good" theory of dévissage for stratified structures. More precisely, what mean the later, i.e., the "dévissage theory for stratified structures"?

Why means concretely means a "dévissage" for a stratification (of a given space say as $X_0 \subset X_1 \subset \dotsb \subset X_n =X$)?

Grothendieck coined the technique of devissage in his dévissage theorem initially in absolute case (i.e. fixed scheme/space $X$). Gruson and Raynaud introduced so called relative dévissage for morphisms $f: X \to S$ (even though — except that the former deals with fixed space $X$, the later with a morphism $f$ — I do not see yet a deeper connection between these two approaches, compare also with this question, especially not the reason to regard the later as extension of the former, but that's another story),
coming back to motivation of this question, what did Grothendieck meant by dévissage for a stratifications?
Can it be explained at least in a hand wavy way with what structure/objects one is dealing with?

Did this considerations involve the relative dévissage concept by Gruson and Raynaud and he meant by this something like glueing dévissages stepwise over strata pieces $X_{i+1} \setminus X_i$ together?

If yes, what kind of "data" should such a dévissages over $i+1$-th piece $ X_{i+1} \setminus X_i$ be considered to be? A criterion (like in the above-linked dévissage theorem for "absolute" case with fixed $X$), a something "like a descent datum" (i.e., like a "glueing plan"), a structure of "atomic" builing blocks, from which one can glue in certain sense all "objects" over strata piece $X_{i+1} \setminus X_i$?

Sorry, if the question is too vague, the problem is really bad intuition with what kind of structure one is dealing with in Grothendieck's vision of dévissage theory for strata pieces.

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  • $\begingroup$ Is your $X_i\backslash X_{i + 1}$ supposed to be $X_{i + 1} \setminus X_i$? The mathematically important difference here is that the $X_i$ and $X_{i + 1}$ seem to be switched, but, in terms of presentation, note the difference between $X_{i + 1} \backslash X_i$ X_{i + 1} \backslash X_i, which does not space as one would probably want, and $X_{i + 1} \setminus X_i$ X_{i + 1} \setminus X_i. $\endgroup$
    – LSpice
    Commented Nov 2, 2023 at 1:38
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    $\begingroup$ @LSpice I think you are right, the strata of a filtration as labelled by the OP are $X_{i+1}\setminus X_i$ $\endgroup$
    – David Roberts ♦
    Commented Nov 2, 2023 at 2:35
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    $\begingroup$ Have you looked at The Stacks Project? I remember early on, one of the high points of The Stacks Project was incorporating Gruson-Raynaud (so that the project unambiguously went “beyond” EGA even in foundations of schemes). $\endgroup$
    – Jason Starr
    Commented Nov 2, 2023 at 10:45
  • $\begingroup$ @LSpice: yes you are right, thanks. The rough idea so far I understand it seems to be to understand the "structure over $X$" stratawise by unductive resonings, namely assume we understood the structure over $X_i$, then to proceed to "next level" we need to know how the structure from $X_i$ attached to to nest piece $X_{i+1}$ and this "attaching informations" seemingly are precisely encoded as" devissage" over $X_{i+1} \backslash X_i$ as "combinatorical datum", but I not see as what this datum should be given... $\endgroup$
    – user267839
    Commented Nov 2, 2023 at 10:51

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