The set-theoretic multiverse as a (bi)category

Question

Joel Hamkin's The set-theoretic multiverse has featured in MO questions before, e.g., here and here. But I was wondering about the best category theoretic angle to take on it.

In the paper Joel writes, rather poetically,

Set theory appears to have discovered an entire cosmos of set-theoretic universes, revealing a category-theoretic nature for the subject, in which the universes are connected by the forcing relation or by large cardinal embeddings in complex commutative diagrams, like constellations filling a dark night sky. (p. 3)

He has given us a couple of kinds of morphism here, but what is the best way to capture this multiverse category theoretically? Which morphisms should we allow?

Is it right to stay at the level of ordinary categories? Since each universe, a model of ZFC, is a category, one might expect the multiverse to be at least a bicategory, as suggested here. Do set theorists consider, say, arrows between two forcing relations between two models?

Answer 1

First of all, thank you very much for the question (the attention given to my multiverse article is flattering). I am keenly interested to hear from the category theorists about this. Meanwhile, allow me to comment from a set-theoretic perspective.

Although set-theorists seldom use category-theoretic terminology and ideas explicitly, nevertheless many of their concerns do have a category-theoretic nature. To give two examples:

• It is a fundamental concern in the theory of large cardinal embeddings to understand when the direct limit of a system of iterations of large cardinal embeddings has a well-founded direct limit, and this issue can be cast category-theoretically as the question of whether a certain category exhibits certain limits.
• The choice of support in an iterated forcing argument, pervasive in set theory, amounts to the use of certain limits in certain categories. For example, the fact that ccc forcing is preserved under finite support iterations can be expressed category-theoretically as the category of ccc forcing notions having direct limits. Other choices of support, such as countable support or Easton support (a mixture of inverse limits and direct limits) or revised countable support, can similarly be cast category-theoretically. Much of our understanding of the power of forcing has come from a detailed analysis of the nature of these different types of iterations.

Even the forcing combinatorics of single forcing notion $\mathbb{P}$, such as the question of closure, chain condition and homogeneity properties, can be cast category-theoretically. Some forcing constructions, such as the stationary tower forcing, combine all the category theoretic ideas above, as the conditions in the forcing involve generic embeddings that are iterated and extended.

In each of these cases, the set-theoretic ideas relate directly to features of the class of all models of set theory that might arise from the construction at hand. In the case of iterated large cardinal embeddings, one is led to consider the models of set theory that arise during the course of the iteration. And in the case of forcing iterations, one of course considers the intermediate forcing extensions that arise from the factors of the forcing iteration. Many set theoretic arguments involve a vast assemblage of intermediate models of set theory connected in a certain precise manner, either by forcing or by large cardinal embeddings, and the analysis of this system is driving the argument. It was in part his kind of situation that I had in mind in the remarks that you quoted.

But another aspect was the observation that set theorists have discovered a huge abundance of models of set theory, with new universes often constructed from known universes in certain precise manners. So the natural inclination when viewing the multiverse as a category, therefore, would be to have an absurdly generous concept, where all the models of set theory appear as the objects, and all of the known ways that they can relate appear as morphisms, including elementary embeddings, the forcing extension relation, embeddings from one model to an inner model of another, the end-extension relation, and so on.

But such an absurd idea, of course, is not how one generally makes progress with category theory. Rather, one wants to choose the objects and morphisms carefully so that the category exhibits desirable features, which can then be fruitfully employed. So, my question for the category theorists would be: what are the category theoretic properties that we might aspire to exhibit in the multiverse? An answer might guide one to a fruitful choice of morphisms.

Answer 2

I am trying to think what would be the starting point and a basic program to "categorize" the multiverse.

Let us begin with the accessible category MOD(ZF) of all models of ZF and usual maps of models. Now, obviously, there is something there that makes it distinguished from other analogous cat of models of first-order theories (say GROUPS).

As Joel has reminded us above, there is a plethora of technologies to build models out of existing ones, either by enlargement (forcing), or by restriction (inner models).

My immediate reaction (from the categorical standpoint) is this:

can we characterize these operations STRUCTURALLY?

I mean: can we, without resorting to the usual set-theoretical details of such constructions, describe them (for instance the various types of forcing enlargements) in a purely diagrammatic way within MOD(ZF)? That would be, or so it seems to me, step one (this step may require to introduce additional maps, in which case it can guide us toward the proper set-up).

Asssuming that one can partially answer in the affermative, step two would be:

axiomatize MOD(ZF) as an accessible category PLUS those structural operations.

Step three would be this: unlike other MOD(T) categories, MOD(ZF) has this fascinating property: given a model M of ZF, ie an element of the cat,

one can onsider the MULTIVERSE of internal models of M (let us call it the chinese boxes principle, if there is no name already).

That seems to suggest, as the PO has in fact hinted at, that MOD(ZF) is in fact a higher cat.

The game is not over. There is a subtle interplay between those internal models and their externalization. Here I have no clue, but it looks like the proper line of thought would be to use the machinery of internal category theory to express this feature.

Step four (excuse me, I am talking free-wheeling) would be: now, given the full axiomatization of Mod(ZF), is there some way to prove that is "unique"? Or are there other models of this axiomatization which are not MOD(ZF)? Most likely there would be other (multi)universes which are not the intended one.

POST SCRIPTUM: I am no expert, so the set-theory adepts should correct me if in the loose sketch of the program above I introduced some unintended nonsense. In particular, I assumed a broad threefold partition of model-building tools, into enlargement, restriction (such as L and variants thereof) and finally internalization (ie starting from a model M create another one which happens to be a set in M). If there is something which does not fit the mentioned labels,add it (them) to the menu. For instance, There are (at least) two types of enlargement, one which keeps the tallness fixed, and another type which makes the model taller. A structural approach to model building inside the multiverse has to account for all those types, so a preliminary accurate taxonomy of basic model building shapes is in order.

Answer 3

To Begin: There are two notions we need to clearly distinguish here: first, the principles which are suspected (or asserted) to hold in the multiverse; and second, the actual intended interpretations of these principles and the implicit bounds placed on them by living in a particular $V$.

First, the principles he discusses are first order, which means they live in the world of math proper. Moreover, they are nice, and in fact there is a rather nice model for them. However, this model and any other such model, can in no way reflect anything other then the first-order principles which are asserted to hold in the multiverse. They are simply objects which exhibit the consistency/coherence of said principles (kind of like exhibiting $\{0\}$ with the operation $\{\langle\langle 0, 0 \rangle,0\rangle \}$ and noting that it satisfies the axioms of a group.)

Second, under the intended interpretation, there can be no actual object which is the multiverse. This follows directly from the Forcing Extension Axiom. The reason being: it is impossible to internally "close off" a forcing notion, because by asserting a particular generic exists, you have just defined how to get around it. More succinctly put, for any separative $\mathbb{P}$, if $G$ is $\mathbb{P}$-generic over $V$, then $G$ is not $\mathbb{P}$-generic over $V[G]$ (since $1 \Vdash \forall \dot{p}\in\check{\mathbb{P}} \,\exists\dot q\in (\check{\mathbb{P}}\backslash \dot{G})(\dot{q}\le\dot{p})$.) Moreover, the "Absorption into L" and "Countability Principle" combine to imply that any $V$ which thinks it has captured the multiverse, is only lying to itself.

My Main Point: The proper multiverse is a flat out, meta object, in the strongest sense possible. The reason for this: you officially cannot get out in front of it, or out run the strength of its intended interpretation (like you can do with inaccessible cardinals and ZFC.)

Addendum: see comments.

Note: If there is an issue please let me know in the comments