Existence of dg realization for 6 functors

Is there a way to lift 6 functors on constructable sheaves to the dg world?

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Welcome to MO. Presumably the answer is "yes". Would you find that an informative answer? -- probably not. I would guess that you also want to learn something a bit more specific about the 6 functor formalism. If that is the indeed the case, then I encourage you to reformulate your question: give a bit more background about what you already know, and what are the things that you'd like to learn from an answer. – André Henriques Jun 15 '12 at 10:34
Welcome to MO, dear David. Dear André, your comment reminded me of a mathematician friend, whose daughter was amazed to see that when she visits other people and in dinner somebody asks "Can you pass me the bread?" the response is passing the bread, and not just saying "yes". – Gil Kalai Jun 16 '12 at 23:16
Ok. Point taken :-). I simply wanted to say that the question could be elaborated a bit, e.g., by mentioning which of the six functors are easy to make dg, and which ones look harder. Or it is maybe the interplay between those functors that is confusing in the dg world?... – André Henriques Jun 17 '12 at 10:06
@Gil Kalai: DG stands for "differential graded". This is a reasonable question for the following reason: the "derived category" of a nice Abelian category, where the "6 functor formalism" lives, is essentially the homotopy-theoretic shadow of the DG-category associated to the Abelian category. But the DG-category itself allows us to make certain desirable constructions--e.g. gluing objects together, which are very difficult in the homotopy-theoretic world. So it would be good to have a lift of our useful functors to the DG world. – Daniel Litt Jun 17 '12 at 15:13
Even formulating the problem for needs in representation theory is non-trivial, e.g. to see that constructible sheaves on $X\times X$ is monoidal under convolution. The problem is how to formulate (upper-*, lower-!) base-change in a homotopy-smart way. Fortunately, this problem has been solved: Francis-Gaitsgory suggest a solution in their paper on chiral algebras using categories of correspondences. The idea is obviously rich enough to carry over to any sheaf theoretic setting. But that format hasn't been published yet, though I understand there is forthcoming work of Gaitsgory-Rozenblyum. – Moosbrugger Jul 11 '12 at 12:53

If I understand correctly your are looking for a dg enhancement of the six operation formalism.

There seem to be a paper that does something very close to it: Yifeng Liu and Weizhe Zheng, Enhanced six operations and base change theorem for sheaves on Artin stacks (available at http://math.columbia.edu/~liuyf/sixi.pdf).

They use the language of $(\infty,1)$-categories, but I think one can adapt it to dg-categories (assuming that one is working over a field of characteristic zero).

EDIT Nov. 27, 2012: the above preprint has been posted on the arXiv: http://arxiv.org/abs/1211.5948

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Has anyone (using whatever language) spelled out the formalism of the four operations for $\mathcal{O}$-modules? Or better yet, for quasi-coherent sheaves? I'd be especially interested in coherence for base-change diagrams. (the only reference I could find is buried in one of Lurie's DAG's and for some reason there is a quasi-affineness condition which I still haven't figured out why appears) – Yosemite Sam Jun 25 '12 at 9:55
Up to now I have only seen enhancements of pull-back and push-forward being discussed (see e.g. arxiv.org/abs/math/0604504 and references threrin, and also Lurie's DAG VIII for the specrtal setting). – DamienC Jun 25 '12 at 14:10
@Sam - you might want to look at Gaitsgory's paper on ind-coherent sheaves.. not precisely what you ask perhaps but explains exactly the formal setting in which such a construction should fit. – David Ben-Zvi Jun 25 '12 at 14:42
I'll have a look at the papers. I think I understand that by some Kan extension nonsense one can has a(n $\infty$) functor from derived stacks to stable categories which to $X$ associates the stable category of quasi-coherent modules (or maybe ind-coherent) and to $X \to Y$ associates the pullback. Again by general colimit nonsense it should follow that all these morphisms have adjoints (which in pathological (non qcqs) cases will be something like pushforward composed with the coherator). What I don't know is how to deal with cartesian diagrams, and given base-change, how to deal with two car – Yosemite Sam Jun 25 '12 at 14:59
tesian diagrams one next to another. There should be some kind of "coherence" result. I have been told that this corresponds to a lift to the category of "correspondences". At this point my comment is getting to vague to make any sense. I think I'll have a look at the papers before yapping any longer. – Yosemite Sam Jun 25 '12 at 15:01

This answer only goes half way, but I think it is worth pointing out that the inner workings of the six functor formalism have been studied a lot in the motivic literature.

Motivic six functor formalisms: The definite reference would be the thesis of Ayoub which you can find on his homepage. The formalism of cross functors (based on unpublished works of Voevodsky and Deligne) gives a general framework how to set up the whole six functor formalism. Another relevant reference is the work of Cisinski-Déglise on triangulated categories of mixed motives. They develop the framework further, with the notions of motivic categories fibered over the category of schemes.

My point is that the input to these frameworks can be things more general than triangulated categories, these frameworks work with stable model categories, $\infty$-categories or dg-categories (as long as you feed in the right data). In the motivic setting, the framework is usually applied to yield six functors for categories of motives. But if you check the validity of the axioms for étale sheaves or $\ell$-adic sheaves, the framework would give you dg-versions of the corresponding derived categories. You should also look at the mixed Weil cohomologies paper of Cisinski-Déglise: plugging in $\ell$-adic cohomology in their constructions gives you dg-versions of $\ell$-adic derived categories.

A grain of salt: The frameworks above do not (explicitly) produce dg-enhancements of the six functors. Four of the functors are easy to deal with because they are derived functors of functors on the level of abelian categories - so they have dg-enhancements. Note, however, the exceptional functors are only constructed on the triangulated level in the abovementioned references. But I think that these can be upgraded to dg-versions: e.g. the construction of $f_!$ via a colimit over the category of compactifications of $f$ and then using the adjoint $i_!$ for an open immersion and $p_\ast$ for a proper map should work in the corresponding dg-setting. Similarly, for the definition of $f^!$ one could apply a version of the $\infty$-categorial adjoint functor theorem, together with suitable compact generation properties. If it's possible, it's only going to be a matter of time before an $\infty$-version of these formalisms will be available in the motivic world.

Constructibility conditions: The frameworks above usually work with fairly big categories. However, in the paper of Cisinski-Déglise, you can find the description of compact objects in these big categories - they agree with the classically defined constructible objects. Moreover, under rather weak assumptions, the six functors also preserve compact objects - the framework above then gives you dg-versions of $\ell$-adic constructible sheaves.

Another grain of salt: Ok, this is all for the algebraic setting, working over schemes and such. If you are more interested in the locally compact topological setting, the literature mentioned above probably does not apply directly. However, all the techniques are there, and I am fairly sure that the framework can be adapted to this setting as well.

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