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.

  • $\begingroup$ Can you briefly expand this idea with $\omega$-categories? $\endgroup$ – Berci Apr 23 '13 at 23:26
  • $\begingroup$ @user18921: Every equivalence relation generalizes equality. That is the whole point of equivalence relations. $\endgroup$ – Asaf Karagila Apr 24 '13 at 6:16
  • $\begingroup$ The fact is that, this idea is developed by people that are already using $\infty$-Categories in their everyday life. But defining properly $\infty$-Categories, or more generally $(\infty,n)$-Categories is a real pain in Set theory/Category theory. On the contrary, if you have a theory of $\infty$-Categories, you have the notion of descrete $\infty$-Categories = Categories, and then the notion of Sets. $\endgroup$ – Damien L Apr 25 '13 at 6:04

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