I'm trying to understand how the six functor philosophy applies to representation theory. Consider the category of classifying stacks $BG$ (assume $G$ discrete for simplicity). To every stack we can assign a triangulated category $D(BG)=D(BG,k):= D(Rep(G))$ the (perhaps bounded) derived category of the abelian category of representations of $G$ (in vector spaces over $k$ algebraically closed of characteristic 0). As far as I understand the formalism of six functors should translate (if at all) as follows:


If $\pi : BG \to pt$ then: 


* $\pi\_{!}(-)$ is a derived version of $(-)^G$ (invariants) = group cohomology.
* $\pi\_\*(-)$ is a derived version of $(-)\_G$ (coinvariants) = group homology .


Generally if $\pi: BH \to BG$ then:


* $\pi\_!(-) = \mathbb{L}Ind^G\_H(-)$,
* $\pi\_\*(-) = \mathbb{R}Coind^G\_H(-)$
* $\pi^\*(-) = \mathbb{L}Res^H\_G(-)$.


I have several questions questions:



> 
> 1. Are there any mistakes/wrong intuitions in the above?
> 2. I'm missing the upper shrieks $\pi^!$ and as such also the duality
>  functor. How do these look in general? What is the "dualizing
>  representation" of a group? Is it something familiar?
> 3. How much of this carries over to the category of algebraic stacks (on the big etale site of schemes) $BG$ for $G$ linear algebraic group? (suppose all this happens over some fixed field for simplicity).
> 
> 
> 



Yoga of six functors for group representations?
The accepted answer here is on a rather negative note -- I don't think that's fair! In fact, I think the correct answer is that all of this works, except that it is $\pi\_\ast$ that gives group cohomology, and $\pi\_!$ that gives group homology.


More precisely, consider the category $C$ of locally compact Hausdorff spaces, say, with the Grothendieck topology of open covers. (One could consider many variants, but this will do.) For any stack $X$ on $C$ (in fact, any sheaf of $\infty$-groupoids), one can define a category $D(X)$, defined via descent. (If $X$ is represented by a locally compact Hausdorff space, this is the category of sheaves with values in $D(k)$, which agrees with the derived category of sheaves of $k$-vector spaces when $X$ is finite-dimensional.) Moreover, $D(X)$ is a closed symmetric monoidal category, and for a morphism $f: X\to Y$ between such stacks, one can define functors $f^\ast$ (with right adjoint $f\_\ast$) and (for a large class of $f$'s) a functor $f\_!$ (with right adjoint $f^!$).


In particular, this applies to $f: BH\to BG$ for discrete $H$ and $G$. In that case, $D(BG)$ is just the derived category of $G$-representations. Moreover any such map $f$ is "etale" and hence $f^\ast=f^!$, both of which agree with restriction. Then $f\_\ast$ is the right adjoint, which is coinduction; while $f\_!$ is the left adjoint, which is induction.


There's probably a better reference for this kind of $6$-functor formalism on topological stacks, but in any case I wrote up something in my notes on [$6$ functors](https://people.mpim-bonn.mpg.de/scholze/SixFunctors.pdf), see in particular Lecture 7 (for the construction of the six functors on locally compact Hausdorff spaces) and the appendix to Lecture 4 (for passing to stacks).


Regarding your last question, yes, everything also works with etale sheaves, and general stacks (as above). But one difference is that in general $D(BG)$ may not be the derived category of its heart.


[Let me note that I strongly disagree with Matthias Wendt here that $B(\mathbb Z/2\mathbb Z)=\mathbb{RP}^\infty$. They are homotopy equivalent, but other than that they are completely different objects, and in particular their $D(X)$'s are rather different, and many of the problems mentioned in that answer are just problems with that approach. It is indeed true that for $f: \mathbb{RP}^\infty\to \ast$ the functors $f\_!$ and $f^!$ are not defined, but for $B(\mathbb Z/2\mathbb Z)\to \ast$ they are.]


