A 2001 paper by F.A. Muller proposes "ARC" (a modified form of Ackermann set theory) as a foundations of mathematics, and argues that it founds category theory more naturally and/or conveniently than other, competing approaches to set theory. Muller's approach looks interesting. I was wondering
- Has it gained many adherents?
- Has it been used as a basis for any foundational works since it was first proposed?
- Have any issues with ARC come to light in the past 12 years?
Nowadays people are getting very excited about an even better approach : the Univalent Foundations Project started by V.Voevodsky.
It is not only including Category Theory but also $\infty$-Category Theory.
In a very sketchy way, the aim is to replace the idea of equality by homotopy which is a more foundational concept.
For your questions :
-I am not able to say if ARC is getting famous or is having a lot of adhrents ; but I can tell you that the Univalent Foundations Project is getting famous and is having a lot of mathematicians working on it
-In 2013, it is clear that such a theory of Cathegory/Class/Set has to include $\infty$-Categories.