DG enhancements of $\ell$-adic derived categories

Question

This question is similar in flavor to Existence of dg realization for 6 functors

Let $X$ be a complex variety and $D(X)$ the bounded derived category of constructible sheaves (the Euclidean topology and field coefficients say) on $X$. Let $\mathcal{L}\in D(X)$. Then there exists a dg-algebra $\mathcal{E}$ whose cohomology is the shifted $Hom$-groups $Hom(L, L[i])$ in $D(X)$ and such that the derived category of (dg-modules of) $\mathcal{E}$ is equivalent to the triangulated subcategory of $D(X)$ generated by $\mathcal{L}$. (Strictly speaking I need some finiteness conditions but I am going to ignore those for now).

Question 1: Does a similar result hold if $D(X)$ is replaced with the corresponding $\ell$-adic `derived category’?

If the answer to 1) is yes, then:

Question 2: Assume that $X$ is defined over a finite field and $\mathcal{L}$ admits a mixed structure (a la Deligne). Then the shifted $Hom$-groups $Hom(\mathcal{L}, \mathcal{L}[i])$ in the ordinary (= non-mixed derived category) inherit mixed structures. Is it possible to lift the mixed structure to the dg-algebra $\mathcal{E}$ produced by an affirmative answer to 1?

In the circles I run in it is standard to assume that 1) is `morally' true but I have no real clue as to what a proof would look like. Not so sure about 2). They are both true for some particular $\mathcal{L}$ (such as if $\mathcal{L}$ is the direct sum of $IC$-complexes corresponding to a stratification by contractible strata). However I am looking for something more general.

A related question:

Question 3: Does the category of all perverse sheaves on $X$ (say in the $\ell$-adic or complex algebraic setting) have enough projectives (or injectives)?

Question 2 has an analogue in the setting of mixed Hodge modules. I do not know the answer there either (except for some very special cases). My reason for asking it in the $\ell$-adic setting is my failure at answering it in the Hodge setting (which I am considerably more familiar with than the $\ell$-adic setup).

Added later: The answer to Question 3 is no in general (look at $Ext$ with skyscrapers).

Added even later: Let me try and explain what 2) is asking for via some "examples". Let $X$ be a complex variety, and $D(X)$ the usual derived category of constructible sheaves. Consider the constant sheaf $\mathcal{L}$ on $X$. Let $\mathcal{E}$ be the complex obtained by applying taking global sections of the Godement resolution of $\mathcal{L}$. Then $\mathcal{E}$ is a dg-algebra. Note:

a) The cohomology algebra of $\mathcal{E}$ is isomorphic to the $Ext$ algebra of $\mathcal{L}$ which is of course the same as the cohomology ring $H^*(X)$.

b) It is not quite obvious, but true (and well known) nonetheless, that the derived category of dg-modules of $\mathcal{E}$ is equivalent to the triangulated subcategory of $D(X)$ generated by $\mathcal{L}$. (To keep things simple I am omitting some finiteness adjectives here).

Now if we had been working with rational coefficients say, then $H^*(X)$ has a lot of additional structure such as a mixed Hodge structure. In particular $H^*(X)$ has an additional grading via weights (if you don't like Hodge theory, feel free to work with Gilet and Soule's motivic weight filtration in which case you can even work over the integers). What Question 2) is asking for is to lift this grading to the complex $\mathcal{E}$ so that the induced grading on $H^*(\mathcal{E})$ is exactly what you started with. Actually, the way it is stated Question 2) is asking to lift the full mixed structure, but for my purposes lifting the grading suffices.

With the Godement resolution it is not clear whether this can be done. However, say instead of the Godement resolution I had used the de Rham resolution (assume $X$ to be smooth), then both a) and b) are still true and now I can lift the weight grading.

My interest is in $\mathcal{L}$ more general than the constant sheaf: mainly some semisimple perverse sheaves. If one is working with the complex numbers, then basically using Godement resolutions (or any injective resolution) one can construct an $\mathcal{E}$ such that a) and b) hold (so this is my dg-algebra model of the category I am after). On the other hand mixed Hodge theory gives me a weight grading on the cohomology of $H^*(\mathcal{E}) = Ext^*(\mathcal{L}, \mathcal{L})$. And Question 2) asks to lift this to $\mathcal{E}$.

Now I simply don't know how to do this in the Hodge setting. But weights in $\ell$-adic cohomology (work over finite fields now) are in a sense much simpler (there's an actual Galois group giving the weights etc.) However, in the $\ell$-adic setup since the triangulated category that one is dealing with isn't quite a derived category, it is not clear if one can construct an $\mathcal{E}$ satisfying a) and b) in the first place. This is what Question 1) is asking to resolve, and then Question 2) is asking to lift weight gradings.

Let me point out that the same questions can be asked in the setting of motivic sheaves (say in the form given by Deglise-Cisinski), and if you believe in "Beilinson's world" of motivic sheaves (spruced up a bit to a dg-enhanced setting) then all of these questions must have positive answers. But the motivic theory as it stands has a deficiency (t-structure!) that makes it even harder to work with it directly for these things.

Finally, I think it's fair to ask why anyone would/should care about this? This is a long story (which I wont attempt to tell here). But it goes back to Deligne-Morgan-Sullivan's landmark result on the formality of the rational homotopy type of smooth projective varieties. These ideas transplanted into representation theory start having some highly non-trivial consequences (well, if one can prove formality). Let me point to:

Formality of classifying spaces

http://arxiv.org/abs/1209.3760

http://arxiv.org/abs/1404.6333

as starting points for why representation theorists might (or should?) care about these things.

Answer 1

Q1. So, I suggest you the following plan of the proof.

1. Note that any Verdier localization of a triangulated category possessing a differential graded enhancement possesses a differential graded enhancement too (at least if you either ignore set-theoretical difficulties or deal with them somehow). The references here are sciencedirect.com/science/article/pii/S0022404997001527 and sciencedirect.com/science/article/pii/S0021869303005829.

2a. Present $D^+(Sh^{et}(X,Ab))$ (the derived category of \'etale sheaves of abelian groups) either as $K^+(Inj\,Sh^{et}(X,Ab))$ or as the localizationof the whole $K^+(Sh^{et}(X,Ab))$.

b. Present the $l$-adic coefficient category as the localization of $D^+(Sh^{et}(X,Ab))$ by $D^+(Sh^{et}(X,\mathbb{Z}[1/l]-\mod))$ using section 5.9 of arxiv.org/pdf/1305.5361.pdf.

Q2a. All the 'mixed' triangulated categories of sheaves you are interested in are rigid. This means that there exist internal Hom's, so that for any two 'mixed' $X,Y$ there is a 'mixed' $Z$ such that the complex computing the morphisms from $X$ into $Y[i]$ is quasi-isomorphic to the one that compute morphisms from $1$ (the tensor unit of the corresponding category; the constant $l$-adic sheaf $\mathbb{Z}_l$ in the etale setting) into $Z$. So, if you want 'weights' on the first complex, it is sufficient to have 'weights on $Z$'.

b. Still, such a $Z$ is only well-defined as an objects of the corresponding derived category. I don't know whether one can endow any differential graded model of the category of mixed sheaves with a mixed structure. This seems to be a difficult question, and I don't think that anybody really studied it.

c. If your $X,Y$ are 'motivic' then $Z$ is motivic also. Then you can apply my theory of weight strutures for motives (that vastly develops the one of Giller and Soule). Note yet that a weight filtration of a (relative) motif $Z$ is described in terms of its certain 'suvmotives' $Z_{w\le i}$ that are not canonically defined (note that weight complexes of Gllet and Soule are only defined up to a homotopy equvalence, whereas $Z_{w\le i}$ correspond to their stupid truncations). The references for my papers are http://arxiv.org/abs/0704.4003, http://arxiv.org/abs/1007.4543, http://arxiv.org/abs/math/0601713, and http://arxiv.org/abs/0903.0091 (these versions are better than the published ones).