The title is vague, my actuall question is the following:
Has the representations of groupoids been systematically studied? Is there any new phenomenon, compare with the representation of groups? (Either brand new things or pitfalls for those who too familiar with representations of groups). Does this point of view simplified any proof of theorems in representation of groups?
I've been only think of this for 15 mins. But I feel like it might be helpful to think of representation of groupoids for the following reasons:
When one talk about local systems, thinking of it as "representation of the fundamental groupoid" seems more natural than talking about "representation of the fundamental group".
When we talk about modules on stacks, if we choose a presentation of the stack (which is a groupoid), can we treat a given module as a representation of the groupoid? (Just as modules on BG gives representations of G.) Studying when two representations give the "same" module will be interesting.