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I am starting something at six operations.
(Do we already have an nLab page on this? I seemed to remember something, but can’t find it.)
In my memory, Lurie points out (and this was known before) that the six operations lift from the derived setup to the stable (infinity,1)-categorical version.
the six operations lift
The reference by May et al that I included in the entry gives the general abstract 1-categorical context for when a caluclus of six operations exists. This should generalize immediately to the analogous $\infty$-categorical contexts.
Grothendieck already had a general setup for six operations, though I do not know how much of general philosophy is actually explicitly published. I think that most influenced work by the general Grothendieck picture is late 1970s work of Mebkhout.
I do not know how much of general philosophy is actually explicitly published
I have never studied this in detail but did try to scan the literature. I had a hard time finding a decent systematic account. Typically dicussions (even talk presentations) about this topic are in the style of “and then we do this and then we have that and now observe this and then we have that” with loads of distracting technicalities about a very specific setup in between. So I was pleased to finally find the May-reference (via an MO reply) which is now the only article that I have seen which at least attempts to give a systematic account of what’s actually going on.
I didn’t try to dig out the ancient references, though, and more than likely did I miss many good contemporary accounts. So I’ll happily take back my above rant if somebody points me to authorative decent accounts.
I have added a tad more information to the Idea-section at six operations. (Not that the entry isn’t still a stub.)
Added today’s arXiv reference
In this paper we develop the formalism of the Grothendieck six operations on o-minimal sheaves. The Grothendieck formalism allows us to obtain o-minimal versions of: (i) derived projection formula; (ii) universal coefficient formula; (iii) derived base change formula; (iv) K"unneth formula; (v) local and global Verdier duality. As an application we show that, in an arbitrary o-minimal structure, the o-minimal sheaf cohomology of a definably connected, definably compact definable group, with coefficients in a field, is a connected, bounded, Hopf algebra of finite type.
I added the reference
where they develop an enhanced version of the six operations formalism for etale cohomology of Artin stacks, using the language of stable (infinity,1)-categories.
Thanks! That’s good to know.
Wolfgang Soergel told us at a conference in Dubrovnik about the neat reinterpretation of the axiomatics of Grothendieck yoga of six functors as a bifibered multicategory formalism; of course usually in derived sense and over correspondences. This is due Hoermann, also working in Freiburg where he chose (multi)derivators for the derived part of the story. I put the links (the first paper and one overview, the second paper is still not out) in the six operations entry.
An approach to six functor formalism as a bifibered multicategory (multiderivator, when appropriate) over correspondences is in
- Fritz Hörmann, Fibered derivators, (co)homological descent, and Grothendieck’s six functors pdf; Fibered Multiderivators and (co)homological descent, arxiv./1505.00974
Looks like there’s some useful information about (co)homology and six operations to extract from Lurie’s MO answer. In what generality can those results be stated?
Hmm, Lurie’s answer is not the answer I would have given. I would have said that if we generalize the interpretations (1) and (3) to $(\infty,1)$-toposes $X$ instead of $\infty$-groupoids, then we identify homology and cohomology with the left and right adjoints of the constant-$\infty$-sheaf functor; but while such a right adjoint always exists, a left adjoint only exists if $X$ is locally contractible, or if we allow it to be a pro-adjoint. In particular, this seems to me an obvious way to define “some version of homology, in terms of sheaf theory, for spaces that are not locally compact”, which Lurie says in a comment to his answer that he doesn’t know how to define. Am I missing something?
Was the request taken to be for something more direct than as the right adjoint of some sheaf functor?
Looking back at #10, two papers by Fritz Hörmann have since appeared, so I added to the references
Six Functor Formalisms and Fibered Multiderivators arXiv:1603.02146; Derivator Six Functor Formalisms — Definition and Construction I arXiv:1701.02152
The structure of the “six operations in Wirthmüller context” is linear dependent type theory: base change adjoint triples compatible with the closed monoidal stucture of the linear types. The way this encodes (twisted, generalized) homology and cohomology, and their duality via Poincaré duality is summarized in the table
twisted generalized cohomology in linear homotopy type theory – table
which comes from the “fact sheet” at Quantization via Linear homotopy types (schreiber).
(This may not capture sheaf-cohomology aspects such as Verdier duality, due to the restriction to “Wirthmüller contexts”. )
Is there any intuition to be had as to when Wirthmüller contexts occur? Looks like someone else is wondering that:
does anyone know a conceptually satisfying description of the difference between a grothendieck context (e.g. six functors in algebraic geometry) and a wirthmuller context (e.g. six functors for local systems of spectra)?
Which results in Marc Hoyois writing
Going back to the shriek thing, the shriek adjunctions on the Wirthmuller context and the Grothendieck context nLab pages do not match the AG use. For instance $f_*$ always has a right adjoint (sometimes denoted by $f^\times$), but it is $f^!$ only if $f$ is proper. Similarly, $f^*$ may have a left adjoint in a variety of situation (sometimes denoted by $f_\sharp$), but it is $f_!$ only if $f$ is etale or something.
but it is $f^!$ only if $f$ is proper.
Which is the case in which one speaks of a Grothendieck context, in which case the notation does match. I think the notation on the page follows that of Fausk-May 05
In what context does $f_*$ always have a right adjoint?
@Mike: In the $D_{QCoh}$ context, $f_*$ has a right adjoint as soon as $f$ is quasi-compact and quasi-separated. Similarly in stable motivic homotopy theory.
@David It looks like you’ve uncovered one of my past misconceptions in this chat thread: homotopy invariant sheaves over a base space are definitely not locally constant.
$f_*$ has a right adjoint as soon as $f$ is quasi-compact and quasi-separated.
Ok. That’s a strange meaning of “always”…
Is there something like a logic of six operations?
The six operations in Wirthmüller form are nothing but the axiomatization of base change/quantification/dependent (co)product in good dependent type theories. e.g. slides 74-84 here
added this pointer:
I am reminded that our entry here is lacking any indication of where Grothendieck would have said all these things attributed to him; a failure, though, that it shares with rest of the literature on the subject that I remember having seen. For instance the above lectures point (exclusively) to Grothendieck’s “The cohomology theory of abstract algebraic varieties” (pdf), but hints of the “yoga” of six operations are visible there only vaguely and using hindsight, it seems.
added these two:
Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (I), Astérisque 314 (2007) [numdam:AST_2007__314__R1_0]
Joseph Ayoub, Les six opérations de Grothendieck et le formalisme des cycles évanescents dans le monde motivique (II), Astérisque 315 (2007) [numdam:AST_2007__315__1_0]
added also these pointers:
Masaki Kashiwara, Pierre Schapira, Chapters II and III of: Sheaves on manifolds, Grundlehren 292, Springer (1990) [doi:10.1007/978-3-662-02661-8]
Marco Volpe, The six operations in topology [arXiv:2110.10212]
Added today’s
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