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Something like
  1. a double-click on the context opens a new window showing all the context's avaiable contents, or
  2. a double-click on the badge opens a new window showing all the items contributing to the badge count

As I think of this, I personally like both options. They'd be convenient shortcuts, and easily dismissed.

One problem is that a double-click in the project sidebar already has a meaning: it focuses on the project. I just discovered this, and I don't care for it at all; it doesn't feel to me like what a double-click should do, though I'm having a hard time saying why.