Dévissage of stratified structures in Grothendieck's "Esquisse d’un programme"

Question

I have a question about the intuition behind Grothendieck's proposed notion of so called "Tame topology" in his Esquisse d’un programme. Grothendieck insisted that theory should admit “dévissage of stratified structures", see e.g. here or here arXiv:1603.03016.

Question: How should one think on intuitive level of this “dévissage of stratified structures"?

The philosophy behind of the concept of usual dévissage I'm familar with deals roughly with deciding when say for a fixed space $X$ and $C(X)$ be an additive category of "nice" objects naturally attached to $X$ (eg is $X$ a scheme, $C(X)$ could be the cat of coherent $O_X$-modules), and $A \subset \text{ob}C(X)$ subset of the objects of $C(X)$ satisfying (context depending additional properties; archetypical property would be eg the $2$-of-$3$-property ) and devissage results are dealing with metastatements when $A$ actually equals $C(X)$.

Now, which intuition one should have for "dévissage of stratified structures"? Say we have a stratified space ( in reasonble sense) $\emptyset =X_{-1} \subset X_0 \subset X_1 \subset ... \subset X_n =X$ and want to build up inductively the knowledge about $X$ from it's strata pieces step by step. So say wlog $X=X_1$ and we "understand" already the pieces $X_0$ and $X_1 - X_0$.

So far I understand the idea & goal of "dévissage of stratified structures" should be results of type to deduce from information from the pieces $X_0$ and $X_1 - X_0$ something about $X (=X_1)$.

But what kind of "information" about $X$ one expects to deduce via this "stratified dévissage"? Say, how would such archetypical metastatement from "dévissage of stratified structures" look like?
Is it of "strong" form (= full amount of information about $X$) in the sense that of we understand $X_0$ and $X_1 - X_0$ + these pieces satisfy certain "dévissage condition" which one might interpret as kind of "glueing data", then it suffice to reconstruct $X$ "totally" up to isomorphy type (that's why "strong" form; the picture I have in mind is eg the Morse theory)

Or does this stratified dévissage works as "weak" form, so in similar vein to the business with $C(X)$ as above? Namely, it not focussed on "reconstructing completely" $X$ as in strong form from strata pieces $X_0$ and $X_1 - X_0$, but just some weaker data functorially associated with $X$ incarnating metastaments more of following kind:

Say we have again as before $C(X)$ (resp. $C(X_0)$ and $C(X_1 - X_0)$) be an additive category of "nice" objects naturally attached to $X$ (resp to $X_0$ and $X_1 - X_0$) and canonical maps $C(X_0) \to C(X) \to C(X_1 - X_0) $ (...so in most reasonable cases $C$ comes functorially, so as presheaf; think of functoriality of $K$-theory with Grothendieck groups).

And my naively guess is that this "weak form" of Grothendieck's "devissage of stratified structures" deals with questions when objects in $C(X)$ can be constructed from objects from $C(X_0)$ and $C(X_1 - X_0)$, so extension problem in terms of homological algebra.

So my question boils down to which form (strong vs weak) Grothendieck refers by "dévissage of stratified structures" in his "Esquisse d’un programme"?

Answer 1

Note that Esquisse d'un programme is a speculative text, and moreover we cannot ask the author for clarification, so the best you can get is a speculative answer.

That said, I do think that Grothendieck intended a strong form of dévissage here: establishing what data is needed to reconstruct $X$ entirely from its strata and suitable glueing data. Some of this data is allowed to be well-defined up to homotopy (which he calls isotopy), which in modern language is easily facilitated by phrasing everything in $\infty$-categorical language.

Let me also point out that the toposic version of this dévissage has been completely worked out, at least for schemes (not generalised to 'tame topology'). Barwick, Glasman, and Haine [BGH18] show that for a qcqs scheme $X$ with a stratification $X \to P$ over a finite poset $P$, the category of étale sheaves (of $\infty$-groupoids) on $X$ can be recovered from those on the strata plus glueing data on certain '(deleted) tubular neighbourhoods' (manifesting in the form of vanishing $\infty$-topoi, after the 1-toposic version developed by Deligne, later revisited by Orgogozo, Gabber, and Illusie [TdG]). They call this recollement instead of dévissage, and the simplest version (for two strata) appears in 8.5.2.

The language in [BGH18] is very abstract, but a lot of the ideas are already present in Travaux de Gabber [TdG] in a more classical language (1-topoi and derived categories). BGH also extract a version for the derived category of abelian sheaves (but I'm not sure there are underived versions, for sheaves of sets or sheaves of abelian groups).

They write about Grothendieck's Esquisse d'un programme in the introduction

"It is not clear to us how much of the work here he anticipated."

To my surprise, Grothendieck wrote that the toposic case was not known (SGA 4 already contains recollement statements, at least for two strata, but I guess they were not yet phrased in terms of vanishing topoi or tubular neighbourhoods). But since §5 of Esquisse d'un programme starts by saying that he's looking for something in a direction quite different from topoi, it seems that Grothendieck was after a more 'topological' rather than 'toposic' statement. Maybe a lift of the dévissage/recollement statements of [BGH18] on $\infty$-topoi to tame spaces. And a version that uses the real/complex topology instead of the étale topology (the 'tubular neighbourhoods' of [BGH18] are only tubular neighbourhoods by analogy; they do not come from actual tubular neighbourhoods).

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References.

[BGH18] C. Barwick, S. Glasman, P. Haine, Exodromy. Preprint/book draft (2018). arXiv: 1807.03281

[TdG] L. Illusie, Y. Laszlo, F. Orgogozo (eds), Travaux de Gabber sur l'uniformisation locale et la cohomologie étale des schémas quasi-excellents. Astérisque 363–364 (2014). DOI:10.24033/ast.935