TITLE: What of the "Sets, Classes, and Categories" approach to the foundations?
QUESTION [11 upvotes]: 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?

REPLY [5 votes]: 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.