What of the "Sets, Classes, and Categories" approach to the foundations?

set-theory foundations category-theory

Solution 1:

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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