My thesis is on Orbispaces.

At this point, I'm not very proud of it. If you just care about generalities on orbispaces, you can view this first part .
It's mostly correct except for a couple of places where there are problems with the small object argument.

The second part deals with the global quotient conjecture. It relies on an obstruction theory that I was not able to build.
I have some ideas on how to fix the whole thing, but if you're looking for a rigorous proof, you'll be disappointed.

I recommend that you start by reading this 4 page extended abstract.