This is closely related to a previous question on the topic, but hopefully adds some motivation.
Let $G_{/\mathbf Q}$ be a semisimple group, $K\subset G(\mathbf R)$ a maximal compact subgroup, and $X=G(\mathbf R)/K$ the associated symmetric space. For $\Gamma\subset G(\mathbf Q)$ a sufficiently small arithmetic subgroup, the action of $\Gamma$ on $X$ is free, so the locally symmetric space $\Gamma\backslash X$ is a $K(\Gamma,1)$-space. If $V$ is a representation of $\Gamma$, we get an induced sheaf $\widetilde V$ on $\Gamma\backslash X$, and there is a natural isomorphism $$ \operatorname H^\bullet(\Gamma\backslash X,\widetilde V) \simeq \operatorname H^\bullet(\Gamma,V) . $$ Now, since $\Gamma\backslash X$ is a topological space, it makes sense to discuss the cohomology with compact supports $\operatorname H^\bullet_\mathrm{c}(\Gamma\backslash X,\widetilde V)$.
Is there a "purely algebraic" description of $\operatorname H^\bullet_\mathrm{c}(\Gamma\backslash X,\widetilde V)$ when $\Gamma$ is an arithmetic group?
[In Joël's question, Brown's book Cohomology of groups is mentioned. It only gives a general definition of $\operatorname H_\mathrm{c}^\bullet(G,\mathbf R)$.]
More generally, if $\Gamma$ is an arbitrary group, we have the classifying topos $\mathcal B(\Gamma)$. Group cohomology $\operatorname{H}^\bullet(\Gamma,V)$ is "cohomology of a topos" $\operatorname H^\bullet(\mathcal B(\Gamma),V)$. For a topos like $\operatorname{Sh}_\mathrm{ét}(X)$ ($X$ a variety) there is a good notion of "cohomology with compact support," or more generally "proper pushforward." I am told there is a notion of "proper morphism" of topoi.
Let $f\colon \mathcal X\to \mathcal Y$ be a geometric morphism of topoi. Is there a reasonable notion of "proper pushforward" $f_!\colon \mathcal X\to \mathcal Y$ (or maybe just $\operatorname{Ab}(\mathcal X)\to \operatorname{Ab}(\mathcal Y)$) that agrees with the requirement $f_!\dashv f^\ast$ when $\mathcal X\to \mathcal Y$ is an open immersion?
