Substructural fixed-point theorems and the diagonal argument: theme and variations

This is just to give a pre-release sneak peek of a pre-preprint I want to put on the arXiv very shortly. It’s a slightly odd beast, since I think it might be of some interest to logicians/philosophers, computer science types and possibly also category theorists, but I can’t tell. It’s a further analysis of Lawvere’s diagonal argument/fixed point theorem, reducing the assumptions beyond what is probably sensible, and giving a few different versions of the fixed point theorem more general than Lawvere’s. Here’s the abstract:

This note generalises Lawvere’s diagonal argument and fixed-point theorem for cartesian categories in several ways. Firstly, by replacing the categorical product with a general, possibly incoherent, magmoidal product with sufficient diagonal arrows. This means that the diagonal argument and fixed-point theorem can be interpreted in some substructural type theories, and semantically in categories with a product functor satisfying no coherence axioms, for instance relevance categories. The second way is by showing that both of Lawvere’s theorems as stated for cartesian categories only concern the well-pointed quotient, and giving a version of the fixed-point theorem in the internal logic of an arbitrary regular category. Lastly, one can give a uniform version of the fixed-point theorem if the magmoidal category has the appropriate endomorphism object, and has a comonad (i.e. an (S4) necessity modality) allowing for potentially `discontinuous’ dependence on the initial data.

Perhaps it is just a bit of a curio, and really waiting for a killer app, but it’s been sitting on my plate for a while, and I think some level of feedback is warranted, since I really can’t tell how it will land with people inside and outside my sphere of expertise. Grab your copy here.

Thuses

(This is a guest post by David Benjamin Lim)

I would like to make an announcement concerning a new math blog, Thuses.com, started by Stanford math graduate students Slava Naprienko, Bogdan Zavyalov and myself. You may already be aware of Mathoverflow and how it has become extremely valuable to the mathematical community. Unfortunately on Mathoverflow you cannot share little things in math – often not enough for a paper – that you find cool with others. 

We feel that the current format in which mathematical ideas are disseminated throughout the community is suboptimal: The arxiv does not support comments and hence is not conducive to discussing mathematical ideas. On the other hand, if you want to publish a “folklore” result, there is not really a place to do this apart from a personal blog or website. It would be nice if there would be a one-stop place to share math, which is what motivated us to start Thuses. 

The things you post on Thuses don’t even need to be new or folklore results. They can be expository in nature, e.g. a new take on a paper or a hairy calculation that’s “known” but no mathematician has ever bothered to write down. Or even a list of challenge problems, such as Peter Scholze’s post on Kevin Buzzard’s blog here. Note however that posts on introductory grad level math, e.g. “Introduction to Scheme Theory.” are not welcome on Thuses. 

For an example of a post that is really awesome, take a look at this post of Sean Cotner and Bogdan. It is exactly the kind of thing that we want to see on Thuses, and answers one of Richard Taylor’s questions during the Stanford Number Theory Learning Seminar.

Come to Thuses and spread the word of math! For more ways to support us, please look at our support page here.

When not to say “universe”

(This post offers no new insights into IUT, it is just an exploration of some of the terminology that gets thrown around. But I had reason to get these ideas down, which I will share later)

Inter-Universal Teichmüller Theory (IUT) is somewhat infamous, to say the least. I want to write about some purely cosmetic thoughts I had recently in a discussion around the use of the term “universe” in IUT. It was curious, because I was, while talking, thinking of the technical term universe i.e. Grothendieck universe and the person that I was talking to had gotten details from a source much closer to the horse’s mouth, and they had the mental image closer to what it apparently really means—and it was not a Grothendieck universe.

However, there seem to be two distinct uses of the term “universe” by Mochizuki. One is the technical sense, given in IUT4 section 3, and explicitly said to be related to size issues and needing to “go up a universe”, as we say, since certain large constructions turn out to stay inside a given Grothendieck universe. I shall always say “Grothendieck universe” for this sense. However, Mochizuki then immediately goes on to say

In the following discussion, it should be understood that every set-theoretic formula that appears is “absolute” in the sense that its validity for a collection of sets contained in some universe relative to the model of set theory determined by is equivalent, for any universe such that , to its validity for the same collection of sets relative to the model of set theory determined by

In plain terms, this means that the definitions in that whole section regarding set-theoretical constructions are not affected by going up a universe. This point is repeated in Remark 3.1.4 of IUT4, which concludes

This is the sense in which we apply the term “inter-universal”. That is to say, “inter-universal geometry” allows one to relate the “geometries” that occur in distinct universes.

But what this means is that given a specific geometric object to start with, which lives (I guess!) in the base universe, does not change in the slightest when thinking of it as living in the next universe up. Talk of computing something in an “alien” universe, for instance as is apparently made possibly by the use of “species”, as in

calculate in terms that make sense in one universe the operations performed in an alien universe!

is odd, since Grothendieck universes are cumulative: given a pair of Grothendieck universes (in standard ZFC set theory) then one contains everything contained in the other (say if is bigger than ). In this sense, “alienness” is not mutual, as from the point of view of the larger Grothendieck universe , everything in just happens to be comparitively rather small, but still lives in V. One could still say that things in are “alien” to things in , but then that would be true of sets and proper classes. No one says proper classes are “alien” to sets. I also note that nowhere is the term “alien” defined, except in a dictionary sense;

Here, the intended sense of the descriptive “alien” is that of its original latin root, i.e., a sense of abstract, tautological “otherness”

no mathematical meaning is attached to it that I’ve seen.

It is also mildly ironic that despite claims that

one must continue to extend the universe, i.e., to modify the model of set theory, relative to which one works. Here, we recall in passing that such “extensions of universe” are possible on account of an existence axiom concerning universes, which is apparently attributed to the “Grothendieck school“

the work of the Grothendieck school, which I take to mean the collected volumes EGA I-IV and SGA I-VII, has recently been proved to not need universes by Colin McLarty (more so because the preprint originally landed before the IUT preprints did, though it only appeared in print last year). McLarty makes the point rather forcefully that Grothendieck was aware of size issues, and that his eponymous universes would ensure nothing ran afoul of Russell’s Paradox, but also that it was merely a technical fix, and not essential. The same point was also emphasised by Brian Conrad (among others: 1, 2, 3) when the issue of Grothendieck universes and the proof of Fermat’s Last Theorem came up.

So despite maybe some technical size issues regarding definable categories (which is what “species” seem to be) in the back end of IUT4, what else gets slapped with the phrase “universe”? Well, there’s this quote from the expository document “The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmüller Theory” (“Alien Copies”):

That is to say, the notion of a “universe”, as well as the use of multiple universes within the discussion of a single set-up in arithmetic geometry, already occurs in the mathematics of the 1960’s, i.e., in the mathematics of Galois categories and étale topoi associated to schemes. On the other hand, in this mathematics of the Grothendieck school, typically one only considers relationships between universes — i.e., between labelling apparatuses for sets — that are induced by morphisms of schemes, i.e., in essence by ring homomorphisms. The most typical example of this sort of situation is the functor between Galois categories of étale coverings induced by a morphism of connected schemes.

So here we see again the invocation of the work of Grothendieck and collaborators (this time, no scare quotes), except “universe” here is not the technical sense, but in a more colloquial sense that a (Grothendieck) topos—or more specifically a Galois category, a special case—is a category with rich enough structure that one can work entirely inside it as if it were the whole mathematical universe. This is nicely encapsulated by a cartoon that Ingo Blechschmidt uses to illustrate some of his work:

The “relationships between universes” mentioned above are geometric morphisms between Grothendieck toposes, these are extremely familiar to working algebraic geometers, and are not seen as somehow dealing with links between “alien universes”. Mochizuki does go on to warn the reader that there are in IUT

much more complicated relationships between the universes […] that are adopted in the Galois categories that occur in the domains and codomains of these links [defined in IUT]

so one should expect some more general kind of morphism between Galois categories. Notice here that he says outright that the “links” (which I assume means the Theta-links and log-links) are relationships between Galois categories. Even allowing for this to perhaps be an slight oversimplification, we are being invited to consider familiar categories, albeit related in a non-standard way, and

it is precisely this sort of situation that is referred to by the term “inter-universal”.

So now we are being explicitly told that the “inter-universal” aspect is not because of needing to pass up to a larger Grothendieck universe because to size issues, and that having absolutely definable categories means structure is preserved under this passage; rather it is about non-standard morphisms between toposes.

And here, one should remember that a Grothendieck universes (with definition explicitly given by Mochizuki) are things defined using the language of ZFC, elementhood and so on, and of course the assumption of their existence is a proper extension of the strength of ZFC. Further, the more Grothendieck universes one assumes exist (one, two, three, 57, countably many, a proper class…), the stronger one’s theory. But a Galois category is something definable in ZFC (and even in weaker systems), and moreover they exist in bucketloads with no strong assumptions whatsoever. Moreover, they are not defined in terms of ZFC, or the language of set theory, but ordinary mathematical objects.

So it seems like there is some collision here between metaphor and precisely-defined mathematics. The technical definition rears its head briefly in “Alien Copies”, namely when in the context of IUT4 a “a given universe, or model of set theory” is mentioned. But we again return to

Recall from the discussion of §2.10 that the notion of a “universe”, as well as the use of multiple universes within the discussion of a single set-up in arithmetic geometry, already occurs in the mathematics of the 1960’s, i.e., in the mathematics of Galois categories and étale topoi associated to schemes [cf. [SGA1], [SGA4]].

It’s entirely possible that “universe” (with quotes) and universe (without quote) really are meant to refer to difference concepts. Such a tiny distinction between naming highly different concepts seems to me to be not exactly the clearest.

So what is my take? Personally, without passing any judgement whatsoever on the actual mathematical content, not to mention the big claim and raison d’etre of IUT, I think using the word “universe” is the colloquial sense should have been avoided. It makes for great publicity, evokes a sense of amazement, but ultimately it is referring to a mathematical setup that is far from novel, and is a terminological clash with Grothendieck universes, which are a technical tool apparently invoked in a different way, but in a way that is not clear exactly how it is used or really needed. Immediately after discussing species and different Grothendieck universes in the introduction to IUT4, Mochizuki says

At a more concrete level, this “inter-universal” contact between constructions in distant models of conventional scheme theory in the log-theta-lattice is realized…

but this latter “inter-universal contact” is nothing other than the generalised kinds of morphisms the “links” between different Galois categories (or their appropriate generalisations) are supposed to represent.

I pretty much think that of the two just-so stories presented above about why the term “inter-universal” is used, it is the second one that is closer to the truth, the one about the “links” being generalised morphisms between more-or-less familiar types of categories. Thus I think from the point of view of trying to communicate this idea, so central as to give the name to the entire topic, something better needs to be said. I think “universe” is too grand a term, and already used in a precise mathematical sense in the community (and in IUT4, too!). In the discussion alluded to at the start, I came up with the metaphorical term of “simulation” to explain in a non-technical way what I think Mochizuki’s “universes” are (i.e. not the precisely-defined Grothendieck universes).

Calling a Galois category a simulation of a particular mathematical object (eg a number field, via its Galois group, hence the Galois category of continuous actions) seems to me to capture the mathematical intuition better. Having independent simulations of isomorphic objects conveys to me that I have a lab and I am looking into little worlds formed out of or around those objects. This is reasonably close to how one might think of a Grothendieck topos more generally, and in fact coincidentally similar to Blechschmidt’s cartoon.

One can go “into” a simulation (in the style of The Matrix, for instance), and see what it looks like from the inside. And maybe one could jump over to a different simulation and see what that process does to anything one tries to carry across from the first simulation. This gels better with the idea that one is comparing independent copies of something, rather than the universe-enlargement idea. Ultimately, one has as many of these little experimental “worlds” as are needed, and they are each “controlled” by a small amount of data. One should note the distinction between ‘big’ and ‘little’ toposes (or ‘gros’ and ‘petit’): the former act like “mathematical universes” (in a sense I don’t want to make precise, but Lawvere’s cohesive toposes are one take on it), the latter act like “generalised spaces”. And it is things like little toposes that I mean when I say “simulation”.

Well, so what? one might say. Well, indeed. Perhaps if there had been less conflation of independent ideas, more clarity around the key concepts, we could have arrived at our current situation sooner. I do not find these conflicting origin stories boost my confidence of the conceptual underpinning of the key mechanisms of the theory, but the reverse. One can explain what I think is supposed to be the key idea (passing between categories arising from arithmetic objects) without so much metaphor. Given that the main audience for IUT should have been arithmetic geometers, writing in terms they are familiar with seems to me to be a given. Metaphors of alien universes being mixed with actual (apparent) use of Grothendieck universes is unneeded. If one wants to tell a story for non-experts, then the metaphors should not outright abuse established terminology, nor outstrip the actual mathematically-precise explanations in volume.

Anyway, I’d be interested to know the impressions people (especially non-experts, and even more, non-mathematicians) have to my very much post-hoc shift in metaphor for this aspect. I think experts got what was meant, and ignored the metaphor. But a lot of pixels have been spilled in the popular media about this whole affair, and I think a better metaphor is needed, if one wishes to discuss it.