[Math] 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.