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Decades ago, Voevodsky constructed the six-functor formalism in motivic homotopy theory [Ayoub's thesis]. This construction seems very technical, long and "hard".

Very recently [Mann's thesis], the six-functor formalism has been defined to be a lax symmetric monoidal functor $D:Corr(C,E)\rightarrow Cat_\infty$ such that the induced functors $\otimes,f^*$ and $f_!$ have right adjoints. This construction is concise and short.

Question: Are the two constructions "equivalent"? Would Mann's definition surprise Voevodsky or would he just say: "this is exactly what I meant."?

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    $\begingroup$ Did Voevodsky in fact have a 6-functor formalism? I was under the impression that this was one of the main new results Ayoub himself proved in his thesis. $\endgroup$ Feb 13, 2023 at 6:08
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    $\begingroup$ My first understanding is that, modulo technical differences between $\infty$-categories and 2-categorical truncations, Ayoub's formalism only needs the data $f^*$, and imposes some form of $\mathbb A^1$-invariance, recovering $f_!$ from some form of Thom isomorphism, but in Mann's formalism, $f_!$'s are separate data. It is also unclear to me how Ayoub's thesis deals with the monoidal structure. $\endgroup$
    – Z. M
    Feb 13, 2023 at 9:00
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    $\begingroup$ @DavidLoeffler, according to a post òd Weibel on AMS, Voevodsky gave a lecture on six functors formalism in 2001-2002, but never published his results. Later Ayoub figured it out and published it in his thesis. $\endgroup$
    – Alexey Do
    Feb 13, 2023 at 13:56
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    $\begingroup$ If you find Ayoub's thesis is incredibly difficult (in fact it is), you may want to take a look at Cisinski book "Triangulated categories of mixed motives" and Ayoub's ICM talk 2014. $\endgroup$
    – Alexey Do
    Feb 13, 2023 at 14:08
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    $\begingroup$ @AlexeyDo The book by Cisinski and Déglise is not independent from Ayoub's thesis: it uses the main result of the latter as a black box. $\endgroup$ Feb 13, 2023 at 21:55

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There may be some confusion in this question about what exactly Voevodsky/Ayoub and Mann do, as they do very different things.

  • Mann's thesis constructs a formalism of six operations in the setting of rigid-analytic geometry, using some $\infty$-categorical construction techniques developed for this purpose by Liu and Zheng. Along the way he gives an abstract definition of what a "formalism of six operations" is using categories of spans, but this definition was certainly well known and already appears (in a more complete form, see below) in the book of Gaitsgory and Rozenblyum.
  • Ayoub's thesis, based on Voevodsky's unpublished ideas, explains how one gets for free a formalism of six operations on the category of schemes out of some very simple axioms (what he calls a "homotopy 2-functor"). These axioms are of a geometric rather than categorical nature. The output of Ayoub's theorem (combined with the $\infty$-categorical construction techniques of Gaitsgory-Rozenblyum or of Liu-Zheng) is in particular a formalism of six operations in the sense of Mann.

Note also that the definition of a formalism of a six operations in Mann's thesis is far from capturing everything. A more complete definition would be a right-lax symmetric monoidal functor from an $(\infty,2)$-category in which both the 1-morphisms and the 2-morphisms are spans (Gaitsgory and Rozenblyum work with an intermediate 2-category, probably for simplicity's sake, in which the 1-morphisms are spans and the 2-morphisms are just morphisms). These 2-morphisms encode the functoriality of bivariant homology, i.e., the isomorphisms $f_!\simeq f_*$ for $f$ proper and $f^!\simeq f^*$ for $f$ étale. But even this does not capture all the structure that we have in the examples, such as the contravariance of bivariant homology with respect to quasi-smooth morphisms.

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    $\begingroup$ Your first bullet is very misleading, it is not accurate at all to say that Mann's definition appears in a more complete form in GR. They require various things to be genuinely symmetric monoidal (as opposed to lax), and they rely on shakily justified (infty,2) stuff. Mann's definition, and the machinery he develops for working with it, is a genuine contribution. $\endgroup$ Feb 14, 2023 at 5:58
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    $\begingroup$ Moreover, in Mann's setup one can actually recover the identifications $f^! =f^*$ for etale $f$ and $f_!=f_*$ for proper $f$, rather than encoding them as additional data. See pp. 40-41 of people.mpim-bonn.mpg.de/scholze/SixFunctors.pdf $\endgroup$ Feb 14, 2023 at 14:10
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    $\begingroup$ @Satan'sMinion Your second point is fair. But I don't agree that my first bullet point is misleading. While Gaitsgory-Rozenblyum only consider a specific example of a formalism of 6 functors, they certainly construct a lax symmetric monoidal functor out of a 2-category of spans, and make it clear that this encodes all the formal relations between the three pairs of functors. The strict monoidality was just a special feature of the example. $\endgroup$ Feb 16, 2023 at 18:47

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