A naive question about representations of group stacks

For an algebraic group $G$ over a field $k$, the abelian category $Rep_k(G)$ of $k$-linear locally finite representations of $G$ can be identified $QCoh(BG_{lis-et})$. Suppose now I replace $G$ with a group stack $\mathfrak{G}$, i.e. a group object in the $\infty$-category of $1$-Artin stacks. Then I can consider the abelian category $D_{qc}(B\mathfrak{G}_{lis-et})^\heartsuit$. Here I am thinking of $B\mathfrak{G}$ as $[ \bullet / \mathfrak{G}]$ which I suppose will be a $2$-Artin stack.

Question: Is there a way to relate $D_{qc}(B\mathfrak{G}_{lis-et})^\heartsuit$ to some category of "representations" of the group stack $\mathfrak{G}$? (Im not even sure what a representation should be maybe something like a functor $B\mathfrak{G} \rightarrow D^{[0 ,1]}(Vect_k)$?)

• Do you want to identify representations with descent data for a vector space along the covering map from Spec $k$ to $BG$? – S. Carnahan Aug 30 '18 at 0:02
• I'm not qualified, but I would guess that the right notion of representation is a representation of the connected components of $\mathfrak G$. – Phil Tosteson Aug 30 '18 at 1:26
• @S.Carnahan Yes I think this is what I want to do. – Anette Aug 30 '18 at 5:35
• I wouldn't expect there to be a lot of interesting reps since they (at least the finite dimensional ones to be safe) will factor through the affinization of your group-stack. Group-stacks do tend to have interesting representations on categories though.. – David Ben-Zvi Aug 30 '18 at 10:50
• Ahh so maybe it would be better to consider a group stack acting on objects of the form $[V_0 / V_1]$ which are obtained by taking the stack quotient of two vector spaces $V_0, V_1$ which are related by a linear map $\varphi: V_1 \rightarrow V_0$? By stack quotient I mean that $\varphi$ defines an action of $V_1$ on $V_0$ by $(v_1, v_0) \mapsto \varphi(v_1) + v_0$. Sorry if that was gibberish. – Anette Aug 30 '18 at 11:28