The n-Category Café

That’s hard. The way I would try (if I was convinced it was possible and I just had to do it) is to make the model of AST from ETCS+R, then somehow use the ZF-algebra structure to figure out what a rank-into-rank embedding looks like. And then…. ?

Thanks, David! That’s kind of you to say so.

I really should add that the second Shulman reference, “Comparing material and structural set theories”, does much more than merely setting terminology, but goes far to address the comparison issues.

Thanks!

It’s a good question.

First, I tend to say things like “traditional, membership-based set theory” rather than the shorter “material set theory” just so as to avoid having to explain what “material” means. I’m not aware of any real difference between the two terms, although there may be some subtleties.

(I remember a well-known material set theorist saying that they didn’t like the term material set theory, because it was only ever used by people who are critical of the subject. That’s probably true.)

Second, I’d say that “categorical set theory” is more specific than “structural set theory”. Mike’s theory SEAR (Sets, Elements, And Relations) is given on the nLab as an example of a set theory that is structural but not categorical. Though maybe it doesn’t quite fit the letter of the definition of “structural set theory” in Mike’s “Comparing material and structural set theories”, which says:

we will use the term […] structural set theory for theories which take functions as fundamental,

whereas SEAR doesn’t literally take functions as fundamental. (It takes relations as fundamental, and a function is a special relation.)

As well as “structural” and “categorical”, there’s “isomorphism-invariant”. And then there’s the not wholly developed notion of neutrality.

Honestly, I probably use these terms more interchangeably than I should, especially when I’m giving a talk or writing something informal on a blog.

As for me, now that I’m mostly into type theory instead, I don’t care so much as I used to about having snappy names for the different kinds of set theory. It would have been nice if “material” had taken off, but nowadays I probably use “membership-based” at least as often myself, for the same reasons. I agree with Tom that they’re basically synonymous.

I also agree that the situation with “structural” and “categorical” is somewhat murkier. I do consider SEAR to be a structural set theory, and I agree that that does seem to contradict the passage Tom quoted from my 2018 paper. However, in that case, the paper even contradicts itself, since a couple of paragraphs later I wrote

Structural set theories … are usually formulated in the language of category theory, [but] there is nothing intrinsically necessary about that (see e.g. [S+09]).

with the citation being to SEAR! (True, the passage Tom quoted was carried over mostly verbatim from the ancestor 2010 paper Stack semantics and the comparison of material and structural set theories — but that’s no excuse, since SEAR dates back to 2009.)

If SEAR is structural but not categorical, then what is the defining feature of a categorical set theory? After all, the sets and functions in any kind of set theory do form a category. Do we require that a “categorical set theory” include the axioms of a category explicitly among its own axioms? If so, then “being categorical” would really be a property of a presentation of a theory.

And, of course, sets and relations also form a category — although the axioms of SEAR don’t explicitly assert this fact, but derive it from a general principle of relational comprehension. I bet one could write down a presentation of ETCS that doesn’t include associativity of composition explicitly, and maybe not even function composition explicitly, instead deriving them from a principle of function comprehension. Would that no longer be a categorical set theory?

Ultimately there’s probably not much point in splitting hairs like this. Clearly ETCS and SEAR belong to the same general circle of ideas and style of theories, at least when compared to ZFC. Sometimes I say things like “structural/categorical set theory” to try to get the advantages of both words.