Can Category theory be founded in set theory using worldly cardinals instead of inaccessibles?

Question

What is exactly demanded for a set theoretic foundation of Category theory? I saw generally two main approaches. One is Muller's who did the work in Ackermann's set theory, but his criteria seem to hint at existence of a universe of $\sf ZFC$ and an $n$-iterative power over it for each natural $n$. It claims that this is all what Category theory needs to be founded in class\set theory. However, many approaches seems to demand having multiple set theoretic universes like Tarski-Grothendieck set theory. Some do it with few inaccessibles.

My question is that why not do all of that in the standard line of set theory using the Worldly cardinals instead.

So for Mullers approach we just need to work in $\sf ZC + \text { a worldly cardinal exists }$, and simply we define Muller's sets as those with ranks lower than the first worldly cardinal.

Now for the multiple universes approach we can also use the worldly cardinals, so we work in $\sf ZFC$ plus a countable sequence $V_{@_1},...,V_{@_w}$ of the first $\omega+1$ stages indexed with worldly cardinals. ["$@_i$" stands for the $i^{th}$ worldly cardinal]. I mean this way we can avoid the superabundancy objections, and also the insufficiency objection, besides we can have a clear cut definition of what the Category of sets is, and define higher Categories as well.

Is this approach possible? What are its pitfalls?

Answer 1

The title asks:

Can Category theory be founded in set theory using worldly cardinals instead of inaccessibles?

The first line in the main body of the question is:

What is exactly demanded for a set theoretic foundation of Category theory?

These are two different questions. As I mentioned in a comment, for the latter question, I find Mike Shulman's article Set theory for category theory highly illuminating. In this answer, let me focus on Muller's paper.

Muller has some idiosyncratic philosophical concerns. He is deeply troubled by strong existence axioms.

The gigantic universe that Feferman creates with ZFCS, that Mac Lane creates with ZFCU, we created with CVN⁺ and Grothendieck with ZFC^ω, is so ridiculously large in comparison to what we actually need to found category-theory, that is unbelievable it has even been considered seriously.

He is under the impression (incorrect IMO) that the motivation for these existence axioms is the belief that they are necessary.

If these proposals are so exessively comprehensive as this author says they are, then why have they been seriously considered in the first place? The answer is simple: because it is generally held that if you want to found category-theory on a set-theory of sorts, then there is no other way but to throw in inaccessibles.

Muller also has what I consider to be an idiosyncratic view of what practicing category theorists need/want.

If the category-theoretician now suddenly wants to have some category of all classes ($\ne \boldsymbol{Cls}$), in the sense of ARC, then we have to kiss her goodbye, for there is no such thing available in ARC. There is however no reason to desire this, because $\boldsymbol{Set}$ and $\boldsymbol{Cls}$ are available as the categories with ‘the least structured objects’, i.e. plain sets or classes of sets as objects and plain functions as arrows.

If you share Muller's beliefs on all three counts, then you might find Muller's proposed solution congenial. However, I would challenge all three assumptions. First of all, inaccessibles or universes are invoked by practitioners primarily for convenience, not because they are believed to be indispensable. When practitioners sense set-theoretic paradoxes crouching at the door, they usually want the simplest possible amulet to ward them off, so that they can go back to thinking about what they really want to think about. Also, especially when it comes to higher category theory, they are often tempted by the thing that prompts Muller to kiss them goodbye—they've set up some framework with (say) two "levels" and they want to be able to invoke all the usual machinery to create a third "level," without having to revise everything they've previously written. Finally, if conservativity over ZFC is the issue, then Feferman's approach has already delivered that; the apparent "superabundancy" can be treated as merely une façon de parler.

To sum up, I would guess that there is nothing seriously wrong with Muller's proposal, but practitioners might find it less convenient than Grothendieck universes, and it's not clear what is really gained. For example, it has long been known that in any particular context where you really want to eliminate the use of universes, then there is no serious obstacle to doing so.

Answer 2

This ultimately depends on what 'kind' of category theory you want to do; it's one of those studies that can really get as 'large' as you want it to, and this is part of what enables us to 'study everything' with it.

Suppose you just want to use category theory to augment your understanding of groups. You can probably get away with just $ZFC$; yes, ${\bf Group}$ is ultimately a locally small category and not a small one, but you can use various tricks (Scott's trick, treating proper classes as their defining sentences in the language of set theory, looking at only groups with certain properties, etc.) to get everything you need in house with $ZFC$.

Now suppose you want to study category theory using category theory. You want to be able to talk about small, locally small, and large categories, and maybe even a few 'levels above this' in a sense that is clear to the categorical intuition but completely undefined in $ZFC$ (outside of absurd stages of 'viewing $n$-classes as their defining formulae', which just feels weird outside of one or maybe two stages to me). You probably could get away with $ZFC$ plus some number of worldly cardinals (corresponding to whatever $n^{th}$ stage of 'class' you need), and this seems (to me) a likely sweet spot for this study. Your suggestion to use the $\omega+1^{th}$ stage is a good one, since $V_{@_\omega}$ is where we would probably want to do $\infty$-category theory for a good ratio of convenience to consistency strength without fussing too much about things that aren't category theory.

Now, suppose you want to study all different kinds of set theories and the relationships between them using category theory. This would require a background theory that allowed you to construct categories of sets in each set theoretical universe, categories of categories in each universe, and categories whose objects are these categories of various kinds from all universes. This is well-defined since (higher) category theory is ultimately the study of structure, and these things have a structure to them, but the foundation involved (if consistent) would have to be able to reproduce models of all possible set theories and thereby prove their consistency, exceeding all of them in consistency strength.

But we're ultimately missing what is probably the most important point here: most working category theorists feel that material set theories are the wrong 'type' of foundation for category theory (see what I did there). If they work with 'set theory' as a foundation at all they usually work in structural set theories, and even this is somewhat unorthodox. The standard answer to the question in the first sentence of your post (with the words set theoretic removed), for most working category theorists, is simply

Type theory.

As someone preferential to material set theories for founding all of mathematics I am sympathetic to questions like this, but I feel it's important to remember that we're an extreme minority in the category theory community -- I suspect that if you polled 100 'working' category theorists on what the proper foundation for category theory is, you would get very few responses that even mentioned set theory at all.