The n-Category Café

This reference, which was “Inspired by Shulman (2008)”, may be of interest:

“Some proposals for the set-theoretic foundations of category theory”, by Lorenzo Malatesta, 2011

If you move fast, you can post a “Large Sets 9.5” on Mahlo and weakly compact sets before I put up Large Sets 10 :-)

There are also lots of sizes above measurable that I’m not going to discuss.

In this series, I’m just talking about the things I happen to find interesting and know (a little) about. That might mean it’s an idiosyncratic selection. Still, it looks like I’m going to get up to 12 posts, which is probably enough.

I wonder if this relates back to the earlier discussion about co-completeness and ETCS. In ZFC, an inaccessible cardinal corresponds to a Grothendieck universe. Thus your question would seem to come down to the difference between a model of ETCS and a Grothendieck universe. And I think that this may have something to do with co-completeness, in the same sense as the earlier discussion.

Thus I’d guess that one might be able to say something like “the category of sets $\lt X$ is a model of ETCS and has (certain) large colimits”. Not sure off the top of my head what the exact statement would be, though, or how that directly relates to the definition of regularity as you gave it.

Yes, I know what you mean. What we want to say is that $X$ is regular iff the category of sets $\lt X$ has $I$-fold coproducts for every $I \lt X$. The obstacle is that in ETCS, we can’t specify an indexed family of sets without specifying its coproduct.

Hmm, the problem with constructing an arbitrary diagram out of $\mathbb{N}$ in ETCS as I had understood it is essentially that ETCS has no set of sets inside it: no ‘universe’ of small sets. But I don’t immediately see that this kind of objection applies here. If we are looking at sets of size less than $X$ for some set $X$, then $X$ plays the role of that ‘universe’.

Indeed, suppose that we wish to construct an $I$-indexed diagram of sets of size less than $X$, where $I$ is any set (viewed as a discrete category). Arbitrary unions of subsets of a given set exist in ETCS, so we can construct the union of the $X_i$’s viewed as subsets of $X \times I$, identifying $X_i$ with $X_i \times \{ i \}$. This union (which is disjoint!) maps canonically to $I$, giving us the desired $I$-indexed diagram. And once we have the $I$-indexed diagram, we can construct the colimit (i.e. coproduct) of it as a set in ETCS (but of course there is no reason to expect it to be a subset of $X$).

Thus, to me, it would seem to make perfect sense to say: $X$ is inaccessible if and only if the following two things hold:

1. the category of sets $\lt X$ is a model of ETCS;

2. for every set $I \lt X$, and every map $d : I \rightarrow 2^{X}$, the coproduct of the sets $d(i)$ for $i \in I$ (which always exists in ETCS for the reason discussed above!) in fact is smaller than $X$.

It could well be that I’m confused about something, but this is the kind of thing I had in mind.

Thanks — I agree with all that. However, it doesn’t answer the question as I intended it, because your condition (2) refers to $X$. When I wrote about filling in the blank here —

• $X$ is inaccessible $\iff$ the category of sets $\lt X$ is/has [SOMETHING]

— I intended “something” to be a property that doesn’t mention $X$, like the other properties in my bulleted list.

The question is whether it’s possible to characterize inaccessibility of $X$ in terms of the abstract category of sets $\lt X$. In other words, if I choose a set $X$ in secret and tell you only the category of sets $\lt X$, up to equivalence, can you figure out whether or not $X$ is inaccessible?

Along the same lines, here’s a completely precise version of the question. Let $\mathcal{S}$ and $\mathcal{T}$ be categories satisfying the ETCS axioms, and let $X \in \mathcal{S}$, $Y \in \mathcal{T}$. Write $\mathcal{S}_{\lt X}$ for the full subcategory of $\mathcal{S}$ consisting of the objects (sets) $\lt X$, and similarly $\mathcal{T}_Y$. Is it true that if $\mathcal{S}_{\lt X} \simeq \mathcal{T}_{\lt Y}$ then $X$ is inaccessible iff $Y$ is inaccessible?

I don’t think inaccessibility of $X$ corresponds to any elementary property of the category of sets $\lt X$. But it can be phrased as a non-elementary property of that category, namely that that category (is a model of ETCS and) has coproducts indexed by its hom-sets. The latter is meant in the usual sense of category theory rather than the internal sense of this category being indexed over itself, so it is a nontrivial property. But it is invariant under equivalence, so it answers this version of the question:

if I choose a set $X$ in secret and tell you only the category of sets $\lt X$, up to equivalence, can you figure out whether or not $X$ is inaccessible?

But it doesn’t answer this version:

Let $\mathcal{S}$ and $\mathcal{T}$ be categories satisfying the ETCS axioms, and let $X \in \mathcal{S}$, $Y \in \mathcal{T}$. Write $\mathcal{S}_{\lt X}$ for the full subcategory of $\mathcal{S}$ consisting of the objects (sets) $\lt X$, and similarly $\mathcal{T}_Y$. Is it true that if $\mathcal{S}_{\lt X} \simeq \mathcal{T}_{\lt Y}$ then $X$ is inaccessible iff $Y$ is inaccessible?

at least, assuming that by “$X$ is inaccessible” you meant “$X$ is inaccessible in $\mathcal{S}$”. If we’re talking about inaccessibility of an object of some model $\mathcal{S}$ of ETCS, then when I said “coproducts indexed by its hom-sets” above that has to be interpreted in terms of $\mathcal{S}$-indexed categories, and so it would only transfer across an equivalence of indexed categories. But $\mathcal{S}_{\lt X}$ is an $\mathcal{S}$-indexed category while $\mathcal{T}_{\lt Y}$ is a $\mathcal{T}$-indexed category, and if $\mathcal{S}\neq \mathcal{T}$ it doesn’t even make sense to ask whether $\mathcal{S}_{\lt X} \simeq \mathcal{T}_{\lt Y}$ as indexed categories.

This is related to the point I made here about first-order versus second-order replacement. If you replace “inaccessible” with “worldly” then the answer to your question is yes, essentially by definition: $X$ is worldly iff $\mathcal{S}_{\lt X}$ is a model of ETCS+R. The extra second-order strength of inaccessibility beyond worldliness refers, I believe irreducibly, to sets outside $\mathcal{S}_{\lt X}$, so I don’t think it can be phrased in terms of elementary properties of $\mathcal{S}_{\lt X}$.

Thanks, that’s helpful.

I don’t think inaccessibility of $X$ corresponds to any elementary property of the category of sets $\lt X$.

OK, good to know that you share that feeling. I wonder whether it can be expressed as a theorem.

I expect we just need to find the right way to phrase the question so that ZFC-theorists can match it up with something they know.