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Given a "nice" scheme over a number field or a finite field, etale cohomology (in part because of its functoriality) provides some powerful invariants of the underlying scheme. Sometimes non-abelian invariants like etale fundamental group are also useful. All of these IIUC factor (in the sense of functors) through the shape of the etale topos. Are there any useful invariants of the etale topos of a scheme that are not completely determined by the shape of the topos?

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    $\begingroup$ Barwick, Glasman, and Haine have constructed a finer invariant that specializes to the shape under a realization called the Galois category, which is naturally stratified over the underlying space of a qcqs scheme. It is a delocalization of the étale homotopy type and determines the higher category of all non-abelian constructible sheaves. They also show that this stratified space determines the étale topos up to equivalence. maths.ed.ac.uk/~cbarwick/papers/exodromy.pdf $\endgroup$
    – Harry Gindi
    Commented Jun 3, 2019 at 12:55
  • $\begingroup$ @HarryGindi I guess one also has to show that it is not determined by the shape of the topos :) $\endgroup$
    – user74900
    Commented Jun 3, 2019 at 13:40
  • $\begingroup$ @cardinal It isn't. There is an equivalence of higher categories proven. All you need to do is cook up two distinct stratified spaces with the same realization with respect to the trivial stratification. $\endgroup$
    – Harry Gindi
    Commented Jun 3, 2019 at 13:57
  • $\begingroup$ @HarryGindi well, maybe consider cooking it up then (I am not saying it is your obligation, your comment is a comment, not an answer), but if you add that information that would make an answer. $\endgroup$
    – user74900
    Commented Jun 3, 2019 at 13:57
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    $\begingroup$ meta.mathoverflow.net/questions/4200/flood-of-similar-new-users $\endgroup$
    – Todd Trimble
    Commented Jun 3, 2019 at 17:29

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As Harry Gindy as said in the comments, there is a refinement of the notion of shape due to Barwick, Glasman and Haine that contains much more information that just the shape. This is not a pro-space, but a pro-(stratified space), and it essentially remembers the Galois groups of every points of your scheme plus how they are glued together along the specialization poset.

The paper is available on the ArXiv as

C. Barwick, S. Glasman, P. Haine, Exodromy, arXiv:1807.03281.

This is maybe a bit unfair to call an "invariant" because it remembers the whole étale topos of a qcqs scheme, thanks to what they call "stratified Hochster duality". Recall that classical Hochster duality tells you that taking the specialization order gives an equivalence between spectral spaces and profinite posets. Then in the exodromyy paper they prove

Theorem 10.10 The functor sending a profinite stratified space $\Pi$ to the ∞-topos of constructible sheaves over $\Pi$ is fully faithful. Moreover the essential image consists exactly of the spectral ∞-topoi.

The paper gives a concrete description of spectral ∞-topoi and even better it proves that the étale ∞-topos of every quasi-compact and quasi-separated scheme is spectral. Thus the profinite stratified shape of a qcqs scheme describes completely the étale ∞-topos.

Even better, the stratified shape of a scheme is always 1-truncated, so you can think of it as a category object in Stone spaces. This is a surprisingly concrete object to manipulate, even if it is not necessarily easy to compute (you need, for example, to know the Galois group of every residue field, and the decomposition group for every specialization).

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