I'm currently writing my master thesis on "Differentiable Stacks". I'm really fascinated by the idea of generalizing manifolds to include also orbifolds/leaf spaces of foliations/moduli spaces...
I formally understand the construction of differentiable stacks as stacks on the category of smooth manifolds possessing a representable epimorphism. And I formally understand why they generalize smooth manifolds.
What I'm having some trouble to get is: how does one come up with such a definition?
I really read many sources on the topic and none of them justifies the choice of such a definition for differentiable stacks (I mean the existence of a representable epimorphism aka an atlas).
This "intuition problem" is getting better as I familiarize more with this object but I feel like I'm just "getting used to it". I'd like to know how this object was born and what's the idea behind it.