The n-Category Café

Most posts in this series introduced some notion of large set. Here’s a diagram showing the corresponding existence conditions, with the weakest conditions at the bottom and the strongest at the top:

Most of these existence conditions come in two flavours. Take beth fixed points, for instance. We could be interested in the literal existence condition “there is at least one beth fixed point”, or the stronger condition “there are unboundedly many beth fixed points” (meaning that whatever set you choose, there’s a beth fixed point bigger than it).

I’ve been a bit sloppy in using phrases like “large set conditions”. These conditions are as much about shape as size. The point is that if a set $X$ is measurable, or inaccessible, or whatever, and $Y$ is another set larger than $X$, then $Y$ need not be measurable, inaccessible, etc. It usually isn’t. But in ordinary language, we’d expect a thing larger than a large thing to be large.

It’s like this: call a building centennial if the number of storeys is a (nonzero) multiple of 100. Even the smallest centennial building has 100 storeys, so all centennial buildings are what most people would call large. In that sense, being centennial is a largeness condition. But a 101-storey building is not centennial, despite being larger than a centennial building, so the adjective “centennial” doesn’t behave like the ordinary English adjective “large”.

Puzzle   There’s one largeness condition in this series that does behave like this: any set larger than one that’s large in this sense is also large. Well, uncountability and being infinite are both conditions with this property, but it’s something less trivial I’m thinking of. What is it?

In the very first post, I wrote this:

I’ll be using a style of set theory that so far I’ve been calling categorical, and which could also be called isomorphism-invariant or structural, but which I really want to call neutral set theory.

What I mean is this. In cooking, a recipe will sometimes tell you to use a neutral cooking oil. This means an oil whose flavour is barely noticeable, unlike, say, sesame oil or coconut oil or extra virgin olive oil. A neutral cooking oil fades into the background, allowing the other flavours to come through.

The kind of set theory I’ll use here, and the language I’ll use to discuss it, is “neutral” in the sense that it’s the language of the large majority of mathematical publications today, especially in more algebraic areas. It’s the language of structures and substructures and quotients and isomorphisms, the lingua franca of algebra. To most mathematicians, it should just fade into the background, allowing the essential points about sets themselves to come through.

And later on in that post, I continued:

While everyone agrees that the elements of a set $X$ are in canonical one-to-one correspondence with the functions $1 \to X$, some people don’t like defining an element of $X$ to literally be a function $1 \to X$. That’s OK. True to the spirit of neutral set theory, I’ll never rely on this definition of element. All we’ll need is that uncontroversial one-to-one correspondence.

I’m not sure I did a great job of explaining what I meant by “neutral”, partly because I was still fine-tuning the idea, but also because it’s one of those things that’s easier to demonstrate than describe.

The proof of the pudding is in the eating. These last dozen posts are a demonstration of what I mean by “neutral”. Throughout, sets have been treated in the same way as a modern mathematician treats any other algebraic or geometric object. In particular, everything was isomorphism-invariant, and we didn’t ask funny questions about sets (like “does $\xi \in x \in X$ imply $\xi \in X$?”) that we wouldn’t ask in group theory if the sets were equipped with group structures, or in differential geometry if the sets had the structure of manifolds, etc.

As promised, I’ve never needed to insist that the elements of a set $X$ literally are the functions $1 \to X$, only that they’re in canonical one-to-one correspondence. Similarly, it’s never been important whether an $I$-indexed family of sets literally is a function into $I$, as long as the two notions are equivalent in the appropriate sense.

Aside   Having said that, I don’t want to concede too much of a point. In modern mathematics, it’s absolutely standard that once you’ve established that two kinds of thing are in canonical one-to-one correspondence, you treat them interchangeably. And sometimes you then rearrange the definitions so that there’s just one kind of thing, making the correspondence invisible. The definition of an element as a map from $1$ is a case of this.

Personally, when I try to follow arguments that an element should not be defined as a map from $1$, I have a hard time finding anything mathematically solid to hold on to. We all agree that the two things are in canonical one-to-one correspondence, so what’s the issue? For comparison, imagine someone arguing that although sequences in a set $X$ correspond canonically to functions $\mathbb{N} \to X$, they shouldn’t be defined as functions $\mathbb{N} \to X$. What would you say in response?

I’m mentioning this not in order to restart arguments that many of us have had many times before (though I know I’m running that risk). My point is that although I’ve scrupulously avoided saying that an element is a function out of $1$, or that an $I$-indexed family is a function into $I$, to do so would actually be pretty neutral in the sense of being mainstream mathematical practice: when two things are equivalent, we treat them as interchangeable.

Let me finish where I started. I began the first post like this:

This is the first of a series of posts on how large cardinals look in categorical set theory.

My primary interest is not actually in large cardinals themselves. What I’m really interested in is exploring the hypothesis that everything in traditional, membership-based set theory that’s relevant to the rest of mathematics can be done smoothly in categorical set theory. I’m not sure this hypothesis is correct (and I suppose no one could ever be sure), which is why I use the words “hypothesis” and “explore”. But I know of no counterexample.

Twelve posts later, I still know of no counterexample. As far as I know, categorical set theory can do everything important that membership-based set theory can do — and not only do it, but do it in a way that’s smooth, natural, and well-motivated.

We could go on with this experiment forever. Personally, I’m very conscious of my own ignorance of set theory. For example, I know almost nothing about two important themes related to large cardinals: critical points of elementary embeddings, and Shelah’s PCF theory. So if the experiment’s going to continue, someone else might have to take the reins.