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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?
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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.

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  • $\begingroup$ Can you briefly expand this idea with $\omega$-categories? $\endgroup$ – Berci Apr 23 '13 at 23:26
  • $\begingroup$ Yes! Please give us more information. Does homotopy generalize equality? $\endgroup$ – goblin Apr 24 '13 at 0:04
  • $\begingroup$ @user18921: Every equivalence relation generalizes equality. That is the whole point of equivalence relations. $\endgroup$ – Asaf Karagila Apr 24 '13 at 6:16
  • $\begingroup$ @AsafKaragila, yes that's a good point. DamienL, if you could give us a bit more information, that would be great. What are the purported advantages of Univalent Foundations? Why might it be more convenient than set theory? And how can you be sure that homotopy is a more foundational concept than equality? $\endgroup$ – goblin Apr 24 '13 at 11:06
  • $\begingroup$ 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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