Fun for Everyone

June 13, 2018

Posted by John Baez

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There’s a been a lot of progress on the ‘field with one element’ since I discussed it back in “week259”. I’ve been starting to learn more about it, and especially its possible connections to the Riemann Hypothesis. This is a great place to start:

Oliver Lorscheid, $\mathbb{F}_1$ for everyone.

Abstract. This text serves as an introduction to $\mathbb{F}_1$ -geometry for the general mathematician. We explain the initial motivations for $\mathbb{F}_1$ -geometry in detail, provide an overview of the different approaches to $\mathbb{F}_1$ and describe the main achievements of the field.

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Lorscheid’s paper describes various approaches. Since I’m hoping the key to $\mathbb{F}_1$ -mathematics is something very simple and natural that we haven’t fully explored yet, I’m especially attracted to these:

Deitmar’s approach, which generalizes algebraic geometry by replacing commutative rings with commutative monoids. A lot of stuff in algebraic geometry, like the ideals and spectra of commutative rings, or the theory of schemes, doesn’t really require the additive aspect of a ring! So, for many purposes we can get away with commutative monoids, where we think of the monoid operation as multiplication. Sometimes it’s good to use commutative monoids equipped with a ‘zero’ element. The main problem is that algebraic geometry without addition seems to be approximately the same as toric geometry — a wonderful subject, but not general enough to handle everything we want from schemes over $\mathbb{F}_1$ .
Toën and Vaquié’s approach, which goes further and replaces commutative rings by commutative monoid objects in symmetric monoidal categories (which work best when they’re complete, cocomplete and cartesian closed). If our symmetric monoidal category is $(\mathbf{AbGp}, \otimes)$ we’re back to commutative rings, if it’s $(\mathbf{Set}, \times)$ we’ve got commutative monoids, but there are plenty of other nice choices: for example if it’s $(\mathbf{CommMon}, \otimes)$ we get commutative rigs, which are awfully nice.

One can also imagine ‘homotopy-coherent’ or ‘ $\infty$ -categorical’ analogues of these two approaches, which might provide a good home for certain ways the sphere spectrum shows up in this business as a substitute for the integers. For example, one could imagine that the ultimate replacement for a commutative ring is an $E_\infty$ algebra inside a symmetric monoidal $(\infty,1)$ -category.

However, it’s not clear to me that homotopical thinking is the main thing we need to penetrate the mysteries of $\mathbb{F}_1$ . There seem to be some other missing ideas….

Lorscheid’s own approach uses ‘blueprints’. A blueprint $(R,S)$ is a commutative rig $R$ equipped with a subset $S \subseteq R$ that’s closed under multiplication, contains $0$ and $1$ , and generates $R$ as a rig.

I have trouble, just on general aesthetic grounds, believing that blueprints are final ultimate solution to the quest for a generalization of commutative rings that can handle the “field with one element”. They just don’t seem ‘god-given’ the way commutative monoids or commutative objects are. But they do various nice things.

Maybe someone has answered this already, since it’s a kind of obvious question:

Question. Is there a symmetric monoidal category $\mathbf{C}$ in which blueprints are the commutative monoid objects?

Maybe something like the category of ‘abelian groups equipped with a set of generators’?

Of course you should want to know what morphisms of blueprints are, because really we should want the category of commutative monoid objects in $\mathbf{C}$ to be equivalent to the category of blueprints. Luckily Lorscheid’s morphisms of blueprints are the obvious thing: a morphism $f : (R,S) \to (R',S')$ is a morphism of commutative rigs $f: R \to R'$ with $f(S) \subseteq S'$ .

Anyway, there’s a lot more to say about $\mathbb{F}_1$ , but Lorscheid’s paper is a great way to get into this subject.

Posted at June 13, 2018 6:17 AM UTC

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With regard to the Riemann hypothesis, the point which seems to me to be crucial is what the analogue of the Frobenius morphism in characteristic $p$ should be. I don’t think any of the approaches to algebraic geometry over $\mathbb{F}_{1}$ which have been tried so far have even any kind of suggestion for this.

Shai Haran’s work (see for example this paper and this one) is that which seems to me to be deepest with regard to this kind of question.

Some people are cynical about the approach to the Riemann hypothesis via $\mathbb{F}_{1}$ . However, it is a truly beautiful picture, and a proof in this way would open up a new kind of mathematics, probably providing a way to make all kinds of dichotomies (e.g. mixed Hodge modules vs perverse sheaves) precise. Thus it is worth striving for.

Posted by: Richard Williamson on June 13, 2018 8:56 AM | Permalink | Reply to this

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Richard wrote:

With regard to the Riemann hypothesis, the point which seems to me to be crucial is what the analogue of the Frobenius morphism in characteristic $p$ should be. I don’t think any of the approaches to algebraic geometry over $\mathbb{F}_{1}$ which have been tried so far have even any kind of suggestion for this.

Thanks! I wondered about that a little, but hadn’t read much about it. I wish people would be more explicit about what they haven’t figured out.

This sounds like a fun bone to gnaw on. It’s a minor miracle that squaring is linear in characteristic 2, etc. It’s less surprising that raising to the first power is linear in characteristic 1. The surprise is that it’s not the identity.

Some people are cynical about the approach to the Riemann hypothesis via $\mathbb{F}_{1}$ .

I’m cynical about lots of things (most of which boil down to “human nature”), but not about the capacity of mathematics to surprise us by being simpler than we expected.

Posted by: John Baez on June 13, 2018 4:50 PM | Permalink | Reply to this

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I am fond of Borger’s suggestion that one should study $\lambda$ -rings as a model for $\mathbb{F}_1$ -algebras. The argument is not that $\lambda$ -rings are $\mathbb{F}_1$ -algebras, but rather that they are $\mathbb{Z}$ -algebras together with descent data along $\mathrm{Spec}(\mathbb{Z}) \to \mathrm{Spec}(\mathbb{F}_1)$ .

Posted by: Theo Johnson-Freyd on June 13, 2018 3:12 PM | Permalink | Reply to this

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There’s a nLab page Borger’s absolute geometry, and a general one for field with one element.

Contributions always welcome.

Posted by: David Corfield on June 13, 2018 3:50 PM | Permalink | Reply to this

Re: Fun for Everyone

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Theo wrote:

I am fond of Borger’s suggestion that one should study $\lambda$ -rings as a model for $\mathbb{F}_1$ -algebras.

I’ve been trying to understand this, so far with little success. Borger’s work is too high-powered for me. More precisely, a bunch of it makes sense, but I don’t get a very basic point: the connection between Frobenius automorphisms and $\lambda$ -rings.

Could someone please say the simplest possible fact in which the words ‘Frobenius’ and ‘ $\lambda$ -ring’ interact in a nontrivial way?

My naive vague idea is something like this. You have a commutative algebra $A$ defined over $\mathbb{Z}$ , otherwise known as a ‘commutative ring’. For each prime $p$ you have some way to lift the Frobenius $F: \mathbb{F}_p \to \mathbb{F}_p$ to a map from $A$ to itself. (I’m not sure I know what that means, and whether you need to consider a choice of all these lifts as extra structure attached to $A$ .) Then these lifts somehow make $A$ into a $\lambda$ -ring. (I don’t know how, and I can’t guess why the lifts for different $p$ would interact in some manageable way, if we’d arbitrarily chosen a lift for each prime $p$ .)

I haven’t found anyone come out and say something like this, though! I know that Grothendieck wrote about $\lambda$ -rings, but I don’t know what he said. The simplest paper I’ve found is

Lars Hesselholt, The big de Rham-Witt complex.

But this is already too hard for me. The simplest passage I can find containing the words ‘Frobenius’ and ‘ $\lambda$ -ring’ is this:

In more detail, let $A$ be a ring, which we always assume to be commutative and unital. The ring $W(A)$ of big Witt vectors in $A$ is equipped with a natural action through ring homomorphisms by the multiplicative monoid $\mathbb{N}$ of positive integers, where the action of $n \in \mathbb{N}$ is given by the $n$ th Frobenius map $F_n : W(A) \to W(A)$ . The Frobenius maps give rise to a natural ring homomorhism $\Delta: W(A) \to W(W(A))$ [….] The triple $(W(A), \Delta, \epsilon$ wiht $\epsilon: W(A) \to A$ the first Witt component is a comonad on the category of rings and a $\lambda$ -ring in the sense of Grothendieck is precisely a coalgebra $(A, \lambda_A)$ of this comonad.

Is this really the simplest thing to say? Do I really need to understand ‘big Witt vectors’ before I can see how Frobenius automorphisms are related to $\lambda$ -rings??? How is a half-Witt like me ever going to penetrate this circle of ideas?

Posted by: John Baez on June 19, 2018 8:32 PM | Permalink | Reply to this

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How is a half-Witt like me ever going to penetrate this circle of ideas?

Oh, you are funny! Seriously, this really made me laugh.

Posted by: Todd Trimble on June 19, 2018 11:55 PM | Permalink | Reply to this

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John wrote

“Could someone please say the simplest possible fact in which the words ‘Frobenius’ and ‘ $\lambda$ -ring’ interact in a nontrivial way?”

Gladly! How about this? “On any $\lambda$ -ring, the Adams operation $\psi_p$ , for any prime $p$ , reduces to the Frobenius map modulo $p$ .”

You didn’t ask for it, but let me say why. I promise to try to keep it short! But I am going to assume you know a bit about $\lambda$ -rings.

We have an identity of symmetric polynomials $x_1^p +\cdots + x_n^p = (x_1+\cdots+x_n)^p + p C(x_1,\dots,x_n)$ where $C(x_1,\dots,x_n)$ is a symmetric polynomial with integral coefficients. In $\lambda$ -ring language, this means we have an identity $\psi_p(x) = x^p +p \delta_p(x)$ on any $\lambda$ -ring $R$ , where $\psi_p$ denotes the $p$ -th Adams operation and where $\delta_p$ is the operation on $\lambda$ -rings corresponding to the symmetric polynomial $C$ above (or rather the compatible family of $C$ ’s as the number $n$ of variables grows). In particular, the $p$ -th Adams operation $\psi_p$ is not just a ring homomorphism but also is congruent modulo $p$ to the $p$ -th power map. In other words, $\psi_p$ is a Frobenius lift. This proves what I said.

The Adams operations also commute. So in fact, in any $\lambda$ -ring, the Adams operations form a commuting family of Frobenius lifts, one for each prime $p$ .

There is also a converse statement: if $R$ is a torsion-free ring equipped with a commuting family of Frobenius lifts $\psi_p$ (again one for each prime $p$ ), then there is a unique $\lambda$ -ring structure on $R$ whose Adams operations are the given Frobenius lifts $\psi_p$ . This is a theorem about symmetric functions, or more precisely about the integrality of the coefficients of certain symmetric functions. I believe it was first proved by Wilkerson (“Lambda-rings, binomial domains, and vector bundles over $CP(\infty)$ ”) and independently by Joyal (“ $\delta$ -anneaux et $\lambda$ -anneaux”). So for torsion-free rings, $\lambda$ -ring structures and commuting families of Frobenius lifts are equivalent concepts.

This is very nice, and we can even make it true for rings with nonzero torsion! The point is that in a $\lambda$ -ring, the congruence between $\psi_p(x)$ and $x^p$ is witnessed (in the words of Rezk) by a particular element $\delta_p(x)$ . So we replace the phrase “Frobenius lift” everywhere with “intelligent Frobenius lift”, where intelligent means that we have to specify witnesses to the Frobenius lift property satisfying their own axioms, reminiscent of how categorification upgrades equations to isomorphisms (i.e. witnesses to the property of being isomorphic) satisfying their own equations. Note that the witness is uniquely determined if our ring is $p$ -torsion free. Then it really is true that a $\lambda$ -ring structure is equivalent to a “commuting family of Frobenius lifts, intelligently understood”. This is also due to Joyal and Wilkerson. The fact that there’s a lot to be gained by working with witnesses to Frobenius lifts, instead of just Frobenius lifts themselves, was independently discovered and pursued by Buium.

Posted by: James Borger on June 20, 2018 12:22 PM | Permalink | Reply to this

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Thanks! This is enough to get me onto the elevator! It’s very clear.

What I really like about it is that it starts with $\lambda$ -rings, which are a structure I have a conceptual grasp of — though I can never remember all the identities — and then asks how they interact with working mod $p$ . You get a commuting family of Frobenius lifts on any $\lambda$ -ring… but you get more, and you accept the fact that you get more, and you work with that.

I’ll have to chew on this for a long time, and learn more about Adams operations and $\lambda$ -rings.

So we replace the phrase “Frobenius lift” everywhere with “intelligent Frobenius lift”, where intelligent means that we have to specify witnesses to the Frobenius lift property satisfying their own axioms, reminiscent of how categorification upgrades equations to isomorphisms (i.e. witnesses to the property of being isomorphic) satisfying their own equations.

Do people ever make that analogy precise? Do they ever do a thing where they take a ring $R$ and a prime $p$ and study the groupoid where the objects are elements of $R$ and a morphism $f: r \to r'$ is an element $a \in R$ such that

$r' = p a + r ?$

This is like an ‘intelligent’ (your words, so nobody blame me) version of $R/p R$ .

Posted by: John Baez on June 20, 2018 4:55 PM | Permalink | Reply to this

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Glad I could help!

As for your groupoid, I’m not aware of anyone working with it in those terms, and I haven’t ever thought of it so explicitly that way myself. But I guess it’s just a disguised form of the complex of abelian groups $R\to R$ , where the map is multiplication by $p$ , right? So if you squint a bit, you might say everyone is doing it, or at least everyone who’s ever considered a complex like that.

Posted by: James Borger on June 21, 2018 9:34 AM | Permalink | Reply to this

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One way to think of that groupoid is to think of $p\times -: R\to R$ as being a presentation, $R[x]\to R$ , with $x\mapsto p.1$ of $R/pR$ as an $R$ -algebra. It then gives a free crossed module internal to the category of commutative rings. I looked at generalisations of this to $R[x_1,\ldots, x_n]$ many years ago and it relates to a simple form of André-Quillen homology and to one of the original forms of cotangent complexes (if I remember rightly).

(Not bothering about whether what follows is really sensible or not) I would add perhaps that the homology of algebraic theories might be of use in some of the Fun stuff to generalise the above. This might really give a higher homotopy slant on Fun perhaps ‘derived fun’!

Posted by: Tim Porter on June 21, 2018 12:16 PM | Permalink | Reply to this

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Maybe you already know, but this is very much like Vitale’s cokernel for Picard categories (grouplike symmetric monoidal groupoids). This cokernel is a symmetric monoidal bicategory whose morphisms are pairs as you’ve suggested, and 2-cells are given by the triangles you would guess.

This specializes nicely to the discrete case (hence a map of abelian groups), and if I recall correctly the construction doesn’t really use invertibility of the objects, except to prove that the cokernel has this same property, hence would work for commutative monoids. There is a long exact sequence for the cokernel which would show, for a ring map f,

the monoid formed by isomorphism-classes of objects is the quotient of f and
the group of automorphisms of the unit is the kernel of f.

BUT this is only a construction of a quotient for a single prime (taking f to be the multiplication by p map). A $\lambda$ -ring has these operations for all primes, so to make the analogy precise I suppose one wants a category with these kinds of morphisms for all primes.

Posted by: Niles Johnson on June 21, 2018 3:24 PM | Permalink | Reply to this

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Ugh; I wrote too quickly and mixed up the two monoid structures a ring has! The multiplication by $p$ map is a homomorphism with respect to the additive structure but not the multiplicative structure! And of course exponentiation by $p$ is the really interesting one, but I should think more before I write more ;)

Posted by: Niles Johnson on June 21, 2018 3:43 PM | Permalink | Reply to this

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James wrote:

But I guess it’s just a disguised form of the complex of abelian groups $R\to R$ , where the map is multiplication by $p$ , right?

Right, that chain complex is just a disguised version of this groupoid. I should have thought a minute more and realized that everyone would think of this as as a chain complex!

So then the question becomes, do people use a chain complex like this for “intelligent Frobenius lifts”?

(I work the other way: I only felt I understood the real point of homological algebra when I learned that nonnegatively graded chain complexes of abelian groups are just strict symmetric monoidal $\infty$ -groupoids. Before that it seemed like a delightful but somewhat arbitrary game.)

(Of course the “just” in the previous paragraph is one of those annoying remarks that category theorists love to make. My student Joe Moeller has dubbed this “justification”. It’s related to categorification, e.g. “the ring of symmetric functions is just the decategorification of the category of Schur functors”.)

Posted by: John Baez on June 21, 2018 5:43 PM | Permalink | Reply to this

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The definition of blueprints piqued my interest because it reminded me of something similar that shows up in a (seemingly) very different context.

Namely, if you restrict to the case that $R$ is a ring, $S$ contains $-1$ , and the nonzero elements of $S$ are invertible, you get what matroid theorists call a partial field. They come up naturally in matroid representability problems, where they essentially serve the purpose of being something “field-like” but with the property that they can have homomorphisms into fields of different characteristics. (Which, I guess, is basically what $\mathbb{F}_1$ is supposed to be about too.)

Partial fields can also be axiomatized, as their name suggests, as something like “fields except the addition can be partially defined”, which perhaps makes them look slightly more “god-given”. If partial fields are to blueprints as fields are to commutative rigs, perhaps blueprints are commutative monoids in “partial commutative monoids”, or something along those lines.

Incidentally, I initially assumed this was basically a coincidence, but thinking about it further it occurred to me that there actually ought to be something to this connection. Matroid representations over fields correspond to points in Grassmannians, so if Lorscheid’s work leads to a good definition of Grassmannians over partial fields there ought to be something to say. Apparently I am not the first to have this thought (see section 9.7).

Posted by: Nick Olson-Harris on June 13, 2018 8:46 PM | Permalink | Reply to this

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