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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJan 10th 2011

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.)

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeJan 11th 2011

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.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJan 11th 2011

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.

• CommentRowNumber4.
• CommentAuthorzskoda
• CommentTimeJan 11th 2011
• (edited Jan 11th 2011)

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.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJan 11th 2011
• (edited Jan 11th 2011)

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.

• CommentRowNumber6.
• CommentAuthorUrs
• CommentTimeJul 7th 2013

I have added a tad more information to the Idea-section at six operations. (Not that the entry isn’t still a stub.)

• CommentRowNumber7.
• CommentAuthorzskoda
• CommentTimeJan 7th 2014

Added today’s arXiv reference

• Mario J. Edmundo, Luca Prelli, The six Grothendieck operations on o-minimal sheaves, arxiv/1401.0846

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.

• CommentRowNumber8.
• CommentAuthoradeelkh
• CommentTimeApr 8th 2014

I added the reference

• Yifeng Liu, Weizhe Zheng, Enhanced six operations and base change theorem for Artin stacks, arXiv.

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.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeApr 8th 2014

Thanks! That’s good to know.

• CommentRowNumber10.
• CommentAuthorzskoda
• CommentTimeJun 28th 2015

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
• CommentRowNumber11.
• CommentAuthorDavid_Corfield
• CommentTimeSep 5th 2017

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?

• CommentRowNumber12.
• CommentAuthorMike Shulman
• CommentTimeSep 5th 2017

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?

• CommentRowNumber13.
• CommentAuthorDavid_Corfield
• CommentTimeSep 6th 2017

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

• CommentRowNumber14.
• CommentAuthorUrs
• CommentTimeSep 6th 2017
• (edited Sep 6th 2017)

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”. )

• CommentRowNumber15.
• CommentAuthorDavid_Corfield
• CommentTimeSep 6th 2017

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.

• CommentRowNumber16.
• CommentAuthorUrs
• CommentTimeSep 6th 2017
• (edited Sep 6th 2017)

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

• CommentRowNumber17.
• CommentAuthorMike Shulman
• CommentTimeSep 6th 2017

In what context does $f_*$ always have a right adjoint?

• CommentRowNumber18.
• CommentAuthorMarc Hoyois
• CommentTimeSep 6th 2017

@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.

• CommentRowNumber19.
• CommentAuthorMike Shulman
• CommentTimeSep 6th 2017

$f_*$ has a right adjoint as soon as $f$ is quasi-compact and quasi-separated.

Ok. That’s a strange meaning of “always”…

• CommentRowNumber20.
• CommentAuthorzskoda
• CommentTimeJan 3rd 2021

The arXiv article of Edmundo-Prelli has been published in the meantime, so the full publication and doi are added.

• CommentRowNumber21.
• CommentAuthorAnton Hilado
• CommentTimeNov 20th 2022

Added a link to Peter Scholze’s lecture notes for his ongoing course on six functor formalisms.

• CommentRowNumber22.
• CommentAuthorAnton Hilado
• CommentTimeNov 20th 2022

Added section on Mann’s approach. Might fill in more later.

• CommentRowNumber23.
• CommentAuthorAnton Hilado
• CommentTimeNov 20th 2022

Added some relevant definitions to Mann’s approach.

• CommentRowNumber24.
• CommentAuthorJ-B Vienney
• CommentTimeNov 25th 2022

Is there something like a logic of six operations?

• CommentRowNumber25.
• CommentAuthorUrs
• CommentTimeNov 25th 2022
• (edited Nov 25th 2022)

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

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