When working with most of modern set theory we tend to transcend ZFC, always working with some strong background hypothesis, whether it being the existence of some elementary embedding, a colouring for some partition property, a generic for some uncountable poset or something completely different. When it comes to using these strong hypotheses in mainstream mathematics it seems that we hit a brick wall, as most of our strong hypotheses don’t easily translate to the language of everyday mathematics.
If we start off by taking a step back and looking at the set-theoretic strong hypotheses with fresh eyes, we see that we’re (at least usually) modelling these assumptions on the existence of numbers. The original idea of an (ordinal) number was that it should be a canonical representative of sets with a certain magnitude — that somehow the number should be a canonical element of the class of all countable sets, for instance. Essentially, we probably could just work with arbitrary countable sets instead of insisting to work with this specific one, but it seems more practical and aesthetic to have fixed this particular representative.
If we now want to say that a category somehow “is” a large cardinal notion, then it seems that we have to let go of this desire to pick canonical representatives, and instead observe what properties a given large cardinal should have, and impose those as conditions on our category. Let’s have a look at (strongly) inaccessible cardinals. These would be sets having the property that
- are non-empty; and
- can’t be approximated from below (i.e. additively and multiplicatively closed as well as having some sort of a regularity property); and
- are closed under taking exponentials.
Imposing these conditions on a category could be to say that
- has an terminal object ; and
- is closed under finite limits and “small” colimits; and
- is closed under taking exponential objects .
Condition (1) is implied by (2), so we can leave that one out, if we prefer. Now, is this sufficient? Would the existence of a category satisfying (1)-(3) be equivalent to the existence of an inaccessible cardinal?
Assuming that we do have an inaccessible , we can form the category , consisting of sets having heriditary size , with functions between them. This category does satisfy conditions (1)-(3) above. Closure under small colimits is granted by regularity of , and exponentials by being a strong limit. But satisfies further properties as well:
- is closed under power sets; and
- has the property that the generator satisfies that any object is bijective to the set of maps .
There’s a category-theoretic notion of a power object, which coincides with power sets when working with a category of sets. The second property is called well-pointedness. So, it’s at least necessary that we should require that our category satisfies these two conditions, on top of the above (1)-(3). A category having all these properties has a name — it’s called a
well-pointed Grothendieck topos.
The question now is whether the existence of a well-pointed Grothendieck topos suffices to yield the existence of an inaccessible cardinal. The apparent obstacle is that we want to convert the algebraic structure into the familiar set theoretic analogues, so that exponential objects really are the familiar set-theoretic exponentials, for instance. This seem to require a functor , preserving all our structure. We could of course instead simply require that is a Grothendieck topos instead of just postulating the existence of some well-pointed Grothendieck topos, which would be equivalent to saying that is inaccessible. But in this case we have to use set theoretic methods to define our topos, so we’re simply using set theory to define a set theoretical notion, defeating the whole purpose of being “independent of set theory”..
..anyway, I better get back to doing some set theory!