In ten words or less: Sets are just 0-categories [7][8]. With more words: Set theory is just 0-category theory. This is obvious today, but two decades ago, Friedman couldn't just go to nCat [0] and educate himself.
Friedman repeatedly demonstrates [1][2] that he isn't interested in grokking why category theory is even a thing; he doesn't see e.g. Pratt's careful explanation that category theory is motivated by studying (natural) transformations. Simpson comes across as a spoiled brat [3] and Friedman comes across as a narcissist who needs to fuel himself by being the bastion of FOM [4]. Seriously, in [5], he has the audacity to simultaneously claim that Lawvere's foundations are the same as Friedman's set-theoretic foundations, and also to ask what a Lawvere theory/sketch is! Unbelievably rude.
I read the entire three-month slapfight again, just to double-check that I hadn't mis-remembered the general outline. Simpson wastes message after message being wrong about Boolean algebras vs. Boolean rings (they're the same picture) and doesn't understand how categorical dualities like Stone duality lead to equivalences. At no point do either of them manage to fully grok a 2-category or how ETCC could be a practical foundations. Tragesser says it well in [5] when he analogizes the entire affair to the sheep and their shop [6], going around and around and always changing the framing but never actually getting to the philosophical meat of the inquiry.
Again, what I see is Friedman and Simpson asking for fairly specific and concrete things, and those things not being provided.
More to the point, what I asked for in my previous comment has also not been provided! What does the knowledge that Sets are "just 0-categories" buy me in concrete terms? Does it facilitate the proof of any theorems in set theory?
Friedman repeatedly demonstrates [1][2] that he isn't interested in grokking why category theory is even a thing; he doesn't see e.g. Pratt's careful explanation that category theory is motivated by studying (natural) transformations. Simpson comes across as a spoiled brat [3] and Friedman comes across as a narcissist who needs to fuel himself by being the bastion of FOM [4]. Seriously, in [5], he has the audacity to simultaneously claim that Lawvere's foundations are the same as Friedman's set-theoretic foundations, and also to ask what a Lawvere theory/sketch is! Unbelievably rude.
I read the entire three-month slapfight again, just to double-check that I hadn't mis-remembered the general outline. Simpson wastes message after message being wrong about Boolean algebras vs. Boolean rings (they're the same picture) and doesn't understand how categorical dualities like Stone duality lead to equivalences. At no point do either of them manage to fully grok a 2-category or how ETCC could be a practical foundations. Tragesser says it well in [5] when he analogizes the entire affair to the sheep and their shop [6], going around and around and always changing the framing but never actually getting to the philosophical meat of the inquiry.
[0] https://ncatlab.org/nlab/show/periodic+table
[1] https://cs.nyu.edu/pipermail/fom/1998-February/001182.html
[2] https://cs.nyu.edu/pipermail/fom/1998-March/001303.html
[3] https://cs.nyu.edu/pipermail/fom/1998-February/001228.html
[4] https://cs.nyu.edu/pipermail/fom/1998-February/001234.html
[5] https://cs.nyu.edu/pipermail/fom/1998-February/001210.html
[6] Edit: I can link directly to the text! https://en.wikisource.org/wiki/Through_the_Looking-Glass,_an...
[7] https://ncatlab.org/nlab/show/0-category
[8] https://ncatlab.org/nlab/show/negative%20thinking