In the recent beautiful talk "Motives and ring stacks" Peter Scholze states the theorem saying that there exists an initial 6-functor formalism on $\mathit{Sch}_\mathbb{Z}$ such that

  • when $f$ is proper then $f^*$ has a right adjoint $f_*$ that commutes with colimits and satisfies base change + projection formula,
  • when $f$ is smooth then $f^*$ has a left adjoint that commutes with limits and satisfies base change and projection formula,
  • and satifies the conditions: excision, $\mathbb{A}^1$-invariance and that the Tate twist is invertible.

Scholze cites Gallauer for this theorem, but wasn't this theorem known to Voevodsky and proven by Ayoub in his thesis already in 2007?

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    $\begingroup$ @LSpice: user.math.uzh.ch/ayoub $\endgroup$
    – M.G.
    Feb 8 at 15:01
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    $\begingroup$ I think that Scholze was referring to Drew–Gallauer. $\endgroup$
    – Z. M
    Feb 8 at 15:11
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    $\begingroup$ @LSpice: Correct me if I'm wrong, but my understanding is that unless someone gets a PhD from a North American institution, there is no particular reason to expect to find their thesis on MathSciNet. $\endgroup$ Feb 8 at 15:20
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    $\begingroup$ Joseph Ayoub's thesis can be found on MathSciNet, it's two Astérisque volumes reviewed as MR2423375 and MR2438151, setting up motivic six-functor formalism for the stable homotopy category. $\endgroup$ Feb 8 at 16:21
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    $\begingroup$ Note that the essential content of Ayoub's theorem is that the properties of $f_*$ in your first bullet point follow from the last two bullet points (actually Ayoub proved it only for projective morphisms, the extension to proper morphisms was done later by Cisinski and Déglise). $\endgroup$ Feb 8 at 22:17

1 Answer 1


My understanding is that Scholze was citing Gallauer for the universal property of the 6FF of motivic spectra (an unpublished result as far as I know), not for the existence of the 6FF. This result was not proved by Ayoub nor was it known to Voevodsky, and it is different from the result of Drew–Gallauer (which is not about 6FFs). The existence of the 6FF is of course due to Joseph Ayoub (in a more general axiomatic context), and was also worked out independently by Oliver Röndigs (unpublished but available online); Voevodsky stated the axiomatic result that Ayoub proves in an unpublished note, but as far as I know there is no record of a proof by him.

ETA: As Peter Scholze explains in the comments, he was in fact referring to Drew and Gallauer during the talk (who prove a universal property for motivic spectra as a lax symmetric monoidal functor on schemes, not on correspondences). In particular, I think there is no claim that motivic spectra satisfy the universal property in Ola Sande's question if "6-functor formalism" is interpreted as a lax symmetric monoidal functor on correspondences.

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    $\begingroup$ Rather that the initial one is given by motivic spectra (the existence of an initial one should be formal). $\endgroup$ Feb 8 at 21:59
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    $\begingroup$ I do not know how to prove it. In fact I am somewhat skeptical of the claim. I could imagine proofs of different claims about 2-categorical 6FFs or with additional axioms about cohomological properness/étaleness of proper morphisms/open immersions. $\endgroup$ Feb 8 at 22:28
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    $\begingroup$ Sorry, I did mean to cite Drew-Gallauer, but more in spirit than in details. When giving my course, I did convince myself that one can prove a precise initiality statement. The claim in the talk was meant to be slightly imprecise as during the discussion I was slightly shifting the intended meaning of 6-functor formalism; in particular, when working with the 2-category, I do not want to enforce a priori a functor from the correspondence category. (Relatedly, a uniqueness conjecture I made in my lecture notes is very likely false. Maybe this is why you are skeptical?) $\endgroup$ Feb 9 at 20:41
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    $\begingroup$ @PeterScholze I see, then I will update my answer. Of course I have no issue with the universal property of Drew-Gallauer. What I am skeptical about is that SH is initial as a functor on correspondences satisfying some axioms that do not clearly force the !-functors to be what they are (which can be done with coh. properness/étaleness axioms or by using a suitable (∞,2)-category of correspondences). If you are referring to the "motivic sheaves" lecture, then I am indeed skeptical of the main theorem there for the same reason (even though it has a different set of axioms than in the question). $\endgroup$ Feb 10 at 6:21
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    $\begingroup$ I agree with the update to the answer. Note also that in the rest of the talk, I was doing the same switch of notion related to what a 6FF is. (Somewhere I mumble something about the functor from Corr(Sch) not being part of the data in the construction of the (\infty,2)-category of motives.) $\endgroup$ Feb 10 at 9:50

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