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Added to F1 a section on Borger’s absolute geometry and then split it off as a stand-alone entry (minimal as it is) Borger’s absolute geometry.
Thanks! And thanks, fixed.
Incidentally, what is this absolute base topos? I.e., what is the site of definition, and what does it classify?
So, from what I can make out, we have a triple of adjoint functors $Symm \dashv U \dashv W: CRing \to \Lambda Ring$, and hence a comonad $U W: CRing \to CRing$. This induces a comonad $Set^{U W}: Set^{CRing} \to Set^{CRing}$, and by one of the results cited in Borger’s second big Witt paper, this restricts to a comonad on $Sh_{Et}(CRing)$. This comonad has a left adjoint given by left Kan extension, and in particular must be left exact. The topos of coalgebras for this left exact comonad on $Sh_{Et}(CRing)$ is what we are calling the absolute base topos. Does that sound right?
Thanks, Todd. We had left this question in the thread on Lambda rings with just the remark that one could try to look at sheaves on rings with Lambda structure, but that instead Borger considers Lambda structures on sheave over rings.
I haven’t had time to do anything further.
Borger seemed to poor cold water in comments on the hope that much could be achieved with his version of F1. Is there reason to be more optimistic?
As I indicated before, I am still wondering if anything singles out the precise details of the construction conceptually, why it should be set in stone. Over in the discussion that you point to my questions are found “sweeping”, and that’s true, but it is unclear to me what it would mean to search for $\mathbb{F}_1$-geometry without cross-checking against the sweeping questions. The whole point of saying “$\mathbb{F}_1$” is that it looks like the right name for the answer to a sweeping question.
If a given proposal for $\mathbb{F}_1$-geometry does not help to explain the function field analogy (instead of just accomodating it), then it is not clear to me exactly why the term “$\mathbb{F}_1$” is supposed to be used.
That said, it does look like “quotienting out the ’Frobenius symmetry’” is a good idea for descending down from $Spec(\mathbb{Z})$. But why not for instance by passing to sheaves on rings with $\Lambda$-structure instead of to sheaves with $\Lambda$-structure on rings? The former seems just as natural and gets us quite close to cohesion, since at least we have an adjoint quadruple connecting $Et(Spec(\mathbb{Z}))$ and $Et(Spec(\mathbb{F}_1))$.
My picture is that there might actually be cohesion and that then the arithmetic fracture square is indeed the left piece of the cohesive fracture square. The support for this idea comes from the nature of the direct image: as Borger highlights, this builds “formal disks” in arithmetic geometry (the “arithmetic jets”) and that is precisley what a cohesive direct image has to do in order for its cohesive fracture to be of the expected integral-adelic kind.
Thanks Urs, this is very nice! Lieven (on G+) already pointed to the connection with the work of Mochizuki. Are the universes in inter-universal Teichmüller theory in any way formally connected to the ones in higher topos theory ?
Dear James,
thanks so much for joining in here. I feel a bit bad now that I fired off half-baked comments here, without quite the leisure to follow up on them right now. I’ll look into this in a bit when I have some time. Am travelling at the moment, with a multitude of distractions.
There are a few recent papers related to toposes and number theory, I thought I’d seen them mentioned somewhere on the nLab, but I cannot currently find the links. The arithmetic site: http://arxiv.org/abs/1405.4527 … as a geometric theory: http://arxiv.org/abs/1406.5479 (also including the topos of cyclic sets and LeBruyn’s topology on the points of the arithmetic site: http://arxiv.org/abs/1407.5538
I am no expert on number theory, but there seem to be some interesting ideas here.
Thanks, Bas, for collecting references. I already ran badly out of time with the loose ends that I introduced so far and should first tie those up before I introduce more. But if you or anyone has time and energy to add this stuff, that would be great.
One quick comment I have is that for the moment I am a little confused by the proposal that the space of points $\mathbb{Q}_+^\times \backslash \mathbb{A}_{\mathbb{Q}} /\hat \mathbb{Z}^\times$ of Connes-Consani/LeBruyn’s “arithmetic site” should give $Spec(\mathbb{Z})/\mathbb{F}_1$: because by the function field analogy via the Weil uniformization theorem this quotient has the interpretation of the space of functions on finitely many pointed formal disks in an algebraic curve, quotiented by the non-vanishing functions on the respective formal disks and on the complement of their puncture points. If one takes the ideles instead of the adeles here, then this is the Cech-representation of the Jacobian of line bundles.
I have tried to come up with some paragraphs that would motivate clearly just why it is, intuitively, that lifts of Frobenius morphisms might be related to descent to $\mathbb{F}_1$. What I have come up with is now at
This expresses thoughts that I have extracted from reading James Borger’s and Alexandru Buium’s articles, but I figured it would help to say it maybe more explicitly, as I tried to do. Or rather, this is what reflects my understanding at the moment, I’d be grateful for complaints by experts, should they be necessary.
For when people get back to an arithmetic mood, in Borger’s absolute geometry it says
induces an adjoint quadruple of functors
but then shows five arrows:
$PSh(Spec(\mathbb{Z})_{et}) \stackrel{\longleftarrow}\stackrel{\longrightarrow}{\stackrel{\longleftarrow}{\stackrel{\longrightarrow}{\longleftarrow}}} PSh(Spec(\mathbb{F}_1)_{et})$
Should the lower one go?
I guess we’d maybe want to take the $Sh$ which comes next as $Sh_{\infty}$. Above in #8, it speaks of cohesion, but I guess to get jet spaces we really need the ’thickening’ of differential cohesion.
If I remember well then the point was that on the level of presheaves there is an adjoint quintuple, but we only know of an adjoint quadruple among these to survive on sheaves.
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