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I know that category theorists have stretched the ontology of collections into conglomerates, 2-conglomerates, etc. My question is how far have they taken this? Are they interested at all in taking this concept as far as it can go, or is it really just a tool for "normal" maths to them? If no one is interested, why not?

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    $\begingroup$ I am not a category theorist, but I have read and done some category theory and never heard of 2-conglomerates. In fact, ACC is the only book where I have ever heard of conglomerates, and it was safe to ignore this concept forever. Better work with universes, if necessary. $\endgroup$ – Martin Brandenburg Dec 23 '13 at 16:17
  • $\begingroup$ One of the things that most interests me is higher category theory (the realm in which I would imagine something like a 2-conglomerate would show up), but I haven't come across this idea before. I'm still, very much, at the beginnings of my studies into this field, so that could be the reason. You might want to take up this question over on the n-category cafe blog, or on nLab: if anyone would know about those kinds of things, they would. $\endgroup$ – user101616 Dec 23 '13 at 16:27
  • $\begingroup$ Martin, I'm sure its safe unless you are interested in certain types of esoterica. Or is it that universes are just as powerful ? I am ignorant on this. Guest, would that necessitate deleting the question here? $\endgroup$ – Jacob Wakem Dec 23 '13 at 16:36
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    $\begingroup$ I dream of a fundamentation where one doesn't have to care about these issues, let $\Bbb{Cat}$ be an object of itself.. $\endgroup$ – Berci Dec 23 '13 at 23:28
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    $\begingroup$ Sufficient large cardinal axioms guarantee you can model transfinitely long successions of conglomerations using larger and larger universes. $\endgroup$ – Kevin Carlson Jan 11 '14 at 0:03
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So this is not a complete answer, but it is an example that I think should make a hierarchy of super-proper classes less interesting.

For any theory $T$ of sets, without proper classes, you can expand it to a theory $T^*$ with proper classes and impredicative class comprehension such that $T^*$ is equiconsistent with with $T$. The proof is due to Hao Wang where historically $T=\mathrm{NF}$ and $T^*=\mathrm{ML}$. If you do this with $T=\mathrm{ZF}$ then $T^*$ is something almost sorta like MK with semisets, which is poorly behaved, so we strengthen $T^*$ with full separation for sets and get, in essence, MK. Now we know MK is roughly equivalent to ZF plus some large cardinal assumptions (which basically give us a Grothendieck universe), so we may as well just use this latter.

The picture I'm trying to draw is this: we can add useless extra layers of classes to an existing theory to no particular benefit, producing $T^*$; or we can strengthen $T^*$ but still have just as much confidence in any strengthening of $T$ that's mutually interpretable with our strengthened $T^*$.

Proof-theoretically, NBG is nice because it's finitely axiomatizable; MK does not share this feature, and I would be very startled if Adamek et al.'s theory did either. Barring this, the only reason I'm aware of to want proper classes and conglomerates is some idea of "naturalness". And while proper classes can (in most treatments) be described solely with $\in$ as those classes that are not $\in$ anything, conglomerates and higher tend to need extra predicates in the language, so it becomes sloppy rather quickly.

Hopefully I've not made any serious errors above; NF/ML are more my area, so I'm largely analogizing using my vague knowledge of ZF relatives. I hope that I am corrected if I have made any serious errors.

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  • $\begingroup$ This seems to be the answer, although I will allow the community a little bit of time to examine it (forgive me, I am an undergraduate with "Wikipedia knowledge" of all of this, though I find it fascinating.) before awarding a bounty. Do you know if we can say call * f, and transfinitely compose f? $\endgroup$ – Jacob Wakem Jan 11 '14 at 2:52
  • $\begingroup$ The construction I'm familiar with doesn't immediately allow for that, but I'm not very well-versed with the model theory behind Wang's result. (I don't hold any advanced degrees either :p) I imagine there's something equivalent by way of large cardinals (as someone mentions in the comments) but I don't know if Wang's method itself can be easily iterated like that. $\endgroup$ – Malice Vidrine Jan 11 '14 at 7:15
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It seems like we can define a power "collection" operator, compose it transfinitely, and apply it to a proper class to expand this notion. Then if we admit class sized ordinals, the fun really takes off! If this attempt fails though, it'd still be useful to document other attempts, if any are out there.

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