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

13$\begingroup$ 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. $\endgroup$– André HenriquesJun 15, 2012 at 10:34

12$\begingroup$ 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". $\endgroup$– Gil KalaiJun 16, 2012 at 23:16

1$\begingroup$ 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?... $\endgroup$– André HenriquesJun 17, 2012 at 10:06

4$\begingroup$ @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 homotopytheoretic shadow of the DGcategory associated to the Abelian category. But the DGcategory itself allows us to make certain desirable constructionse.g. gluing objects together, which are very difficult in the homotopytheoretic world. So it would be good to have a lift of our useful functors to the DG world. $\endgroup$– Daniel LittJun 17, 2012 at 15:13

2$\begingroup$ Even formulating the problem for needs in representation theory is nontrivial, e.g. to see that constructible sheaves on $X\times X$ is monoidal under convolution. The problem is how to formulate (upper*, lower!) basechange in a homotopysmart way. Fortunately, this problem has been solved: FrancisGaitsgory 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 GaitsgoryRozenblyum. $\endgroup$– MoosbruggerJul 11, 2012 at 12:53
2 Answers
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 dgcategories (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

$\begingroup$ Has anyone (using whatever language) spelled out the formalism of the four operations for $\mathcal{O}$modules? Or better yet, for quasicoherent sheaves? I'd be especially interested in coherence for basechange diagrams. (the only reference I could find is buried in one of Lurie's DAG's and for some reason there is a quasiaffineness condition which I still haven't figured out why appears) $\endgroup$ Jun 25, 2012 at 9:55

$\begingroup$ Up to now I have only seen enhancements of pullback and pushforward being discussed (see e.g. arxiv.org/abs/math/0604504 and references threrin, and also Lurie's DAG VIII for the specrtal setting). $\endgroup$– DamienCJun 25, 2012 at 14:10

$\begingroup$ @Sam  you might want to look at Gaitsgory's paper on indcoherent sheaves.. not precisely what you ask perhaps but explains exactly the formal setting in which such a construction should fit. $\endgroup$ Jun 25, 2012 at 14:42

$\begingroup$ 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 quasicoherent modules (or maybe indcoherent) 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 basechange, how to deal with two car $\endgroup$ Jun 25, 2012 at 14:59

$\begingroup$ 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. $\endgroup$ Jun 25, 2012 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 CisinskiDéglise 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 dgcategories (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 dgversions of the corresponding derived categories. You should also look at the mixed Weil cohomologies paper of CisinskiDéglise: plugging in $\ell$adic cohomology in their constructions gives you dgversions of $\ell$adic derived categories.
A grain of salt: The frameworks above do not (explicitly) produce dgenhancements 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 dgenhancements. 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 dgversions: 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 dgsetting. 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 CisinskiDéglise, 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 dgversions 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.