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Martin Brandenburg (Apr 15 2026 at 17:31):I wrote up a convenient set-theoretic foundation for category theory. I first developed it for the third edition of my textbook (still in progress), then for a short reference paper (also in progress), and now for CatDat (live). You can read it here:
I would appreciate your feedback. In my experience, this approach has eliminated many so-called "set-theoretic issues" when revising my book, nothing feels unclear anymore, and the theory has become much more flexible.
Specific questions:
1) Is there a more established source where that approach is used? I know it is just "ZFC + two Grothendieck universes" and "categories are not locally small by default" (!), with the distinction that I use the terms sets, collections, and hypercollections.
2) Do you see any mathematical disadvantages of this foundation? (Let's ignore the 1% of cases where we need more levels of size.)
3) Do you see any pedagogical disadvantages, especially in the context of a textbook? (I introduce categories intuitively long before bringing in Grothendieck universes, which seems important. Also, the textbook section explains Grothendieck universes in much more detail.)
Ralph Sarkis (Apr 15 2026 at 17:41):Only took a quick look. Is it the same thing as sets, classes, and conglomerates from The Joy of Cats (Sections 2.1--2.3) ?
Jonas Frey (Apr 16 2026 at 08:14):I think the classical axiomatizations with classes and conglomerates have the disadvantage that eg the category of classes is not cartesian closed. Using Grothendieck universes, the larger levels have all the closure properties one would like.
Martin Brandenburg (Apr 16 2026 at 14:46):Ralph Sarkis said:
Only took a quick look. Is it the same thing as sets, classes, and conglomerates from The Joy of Cats (Sections 2.1--2.3) ?
It is fundamentally different. Classes are not adequate for category theory (which is weird since so many books and texts are using it).
Kevin Carlson (Apr 16 2026 at 18:27):I think careful people using classes spend (waste?) a lot of of energy avoiding constructing things like the category of all endofunctors of Set, rather working with just a few endofunctors at a time, while less careful people are basically working in your foundation even if they claim to be using classes.