Set-theoretical multiverses and their representation as functors? Why *the* multiverse? In some related MO questions like The set-theoretic multiverse as a (bi)category it is discussed how one might represent the multiverse (see  The set-theoretic multiverse) in a category theoretic way, e.g. by a category having universes as objects (models of ZFC). However it seems not so obvious which morphisms to take for such an approach (e.g. forcing relation or large cardinal embeddings or any other techniques producing new models from old ones). 

Question 1. What implications do the different choices for a method to produce new set theoretic worlds from old ones have w.r.t. the multiverse / is there a right choice (what are the criteria)? If not, then the unfinished discussion about the right morphisms in a category theoretic representaation of a multiverse seems to just reflect this?
Question 2. However, is it essential in this context to talk about the multiverse rather than a multiverse? If it is not obvious which morphisms could be best, could't one regard different multiverses or multiverse representations and treat them equally until one finds justification to single out some i.e. the real multiverse? 
Question 3. Couldn't it be the better way to think of a multiverse as a functor $$V: T\rightarrow M$$ mapping extensions of ZFC in $T$ to ZFC-models in $M$, after all this seems to be natural, since a multiverse is about variable worlds (and then think about multiverses with certain properties like having a left adjoint $V_*:M\rightarrow T$ or to study the functor category of multiverses $M^T$ or the like)?  

 A: Of course we have been investigating a wide variety of
multiverse concepts, and in this sense, yes, we have not just one,
but many, multiverses.
But to be sure, much of this multiverse analysis has been inspired by
the philosophical idea that there is or might be a unique grand
multiverse for mathematics: the actual multiverse, but aware of the possibility that this is a mirage. Even when one has the idea, as many of us do, of looking for mathematical
truth in various set-theoretic universes or categories, there might
remain the question: which such universe or categories are there?
What kind of universes can there be? To assert that there are
facts-of-the-matter about what kind of universes exist is to take a
step in the direction of the multiverse, and this is philosophical issue at stake in the singularity or plurality of the multiverse. Of course it is not satisfactorily answered simply by adopting the multiverse position itself. 
I was led in my multiverse paper, to which you have linked, by this kind of
thinking to formulate axioms expressing what kind of existence
principles we would want or expect in such a multiverse.


*

*Hamkins, Joel David, The set-theoretic multiverse, Rev. Symb. Log. 5, No. 3, 416-449 (2012). DOI:10.1017/S1755020311000359, ZBL1260.03103, blog post
How nice it was to observe that the philosophical issues transform into purely mathematical
questions, when one proposes various specific mathematical
multiverse conceptions and begins to analyze their mathematical
nature. Thus, one gains philosophical insight by means of a purely
mathematical investigation. And this investigation has been
undertaken in earnest.
So the basic approach has been to study various specific multiverse
conceptions as toy multiverses of a kind, standing in for the
actual larger multiverse that we seek to understand.
Let me mention several instances.
Victoria Gitman and I looked at the multiverse consisting of the
countable computably saturated models of set theory, in our paper:


*

*Gitman, Victoria; Hamkins, Joel David, A natural model of the multiverse axioms, Notre Dame J. Formal Logic 51, No. 4, 475-484 (2010). DOI:10.1215/00294527-2010-030), ZBL1214.03035, blog post.


This collection of models, it turns out, fulfill all of the
multiverse axioms I had identified in my multiverse paper, and
more.
Woodin had defined the generic multiverse of a model of set
theory, which is the smallest collection containing that model and
closed under forcing extensions and ground models. This is, of
course, a robust multiverse conception, but nevertheless, there are
many set-theoretic principles, such as the GCH, that are achievable
over a model of set theory, but not necessarily by set forcing.
Benedikt Löwe and I had investigated the generic multiverse as
a Kripke model with two natural modal operators, an upward-oriented
forcing possibility and a downward-oriented ground-model
possibility, and we explored the modal validities of this system in
our papers on the modal logic of forcing and also in:


*

*Hamkins, Joel David; Löwe, Benedikt, Moving up and down in the generic multiverse, Lodaya, Kamal (ed.), Logic and its applications. 5th Indian conference, ICLA 2013, Chennai, India, January 10--12, 2013, Proceedings. Berlin: Springer (ISBN 978-3-642-36038-1/pbk). Lecture Notes in Computer Science 7750, 139-147 (2013). DOI:10.1007/978-3-642-36039-8_13, ZBL1303.03078, blog post.


The Vienna approach, advanced by Sy Friedman and his collaborators, including Carolin Antos and others, has focused
on the collection of countable transitive models of ZFC, the set-theoretic 
hyperverse, which is a multiverse conception, in
which the concept of well-foundedness is emphasized as absolute.
Friedman has identified a program of using the hyperverse as a
means of identifying natural set-theoretic axioms, such as the
Inner Model Hypothesis.
The New York approach, in contrast, in work of myself, Victoria Gitman and Kameryn Williams, has emphasized the non-absolute character of well-foundedness, and has accordingly accommodated ill-founded models of set theory. There have been many recent developments in this area, including the universal finite
set:


*

*J. D. Hamkins and W. Woodin, The universal finite set, ArXiv:1711.07952, pp. 1-16, 2017. blog post
Recently, in my work with Øystein Linnebo and others, we have
explored diverse concepts of set-theoretic potentialism, which I
view as multiverse perspectives. 


*

*J. D. Hamkins and Ø. Linnebo, The modal logic of set-theoretic potentialism and the potentialist maximality principles, to appear in Review of Symbolic Logic, 2018. Arxiv:1801.04599, blog post
In that paper, we look at various multiverse conceptions of potentialism,
exploring collections of models of set theory under the relation of
rank-extension and end-extension and many others, investigating in each
case the modal validities of that potentialist conception.
In summary, there is a huge effort currently underway to
investigate diverse multiverse conceptions.
Lastly, I notice that you mention the category-theoretic approach to the multiverse, so let me mention that, unfortunately, the category-theoretic perspective has not seemed to play a large role in recent work in this area. Perhaps ideas arising from this perspective will find a fruitful application in the future. I am hopeful about that.
