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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJul 23rd 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexAdded to _[[F1]]_ a section _[on Borger's absolute geometry](http://ncatlab.org/nlab/show/field+with+one+element#BorgerAbsoluteGeometry)_ and then split it off as a stand-alone entry (minimal as it is) _[[Borger's absolute geometry]]_.

   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.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeJul 25th 2014
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   Author: zskoda
   Format: TextGreat entry. I remember when Manin got interested in Borger's idea around 2008 (before the preprint, I think). Quadruple of functions should be corrected to quadruple of functors.

   Great entry. I remember when Manin got interested in Borger's idea around 2008 (before the preprint, I think). Quadruple of functions should be corrected to quadruple of functors.
3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJul 25th 2014
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   Author: Urs
   Format: MarkdownItexThanks! And thanks, fixed.

   Thanks! And thanks, fixed.

4. • CommentRowNumber4.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 25th 2014
   • PermaLink
   Author: Todd_Trimble
   Format: MarkdownItexIncidentally, what *is* this absolute base topos? I.e., what is the site of definition, and what does it classify?

   Incidentally, what is this absolute base topos? I.e., what is the site of definition, and what does it classify?

5. • CommentRowNumber5.
   • CommentAuthorTodd_Trimble
   • CommentTimeJul 26th 2014
   • (edited Jul 26th 2014)
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   Author: Todd_Trimble
   Format: MarkdownItexSo, 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?

   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?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJul 26th 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks, 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.

   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.

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeJul 27th 2014
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   Author: David_Corfield
   Format: MarkdownItexBorger seemed to poor cold water in [comments](http://mathoverflow.net/questions/176798/function-field-analogy-and-global-absolute-geometry) on the hope that much could be achieved with his version of F1. Is there reason to be more optimistic?

   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?

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeJul 27th 2014
   • (edited Jul 27th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexAs 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.

   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.

9. • CommentRowNumber9.
   • CommentAuthorJames Borger
   • CommentTimeJul 27th 2014
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   Author: James Borger
   Format: TextHi everyone, I am delighted to see the discussion here and thought I'd jump in and try to help to clear up a few things. One question was about the difference between (1) sheaves on lambda-rings-opposite and (2) lambda-structures on sheaves on rings-opposite (in the etale topology, say, but that's not too important). The two are different but only for essentially technical reasons, and with a bit of tweaking, they can be made the same. Let me explain. First, lambda-rings and Witt vectors are all about commuting lifts of Frobenius maps on rings (commutative), but this particular issue has nothing to do with Frobenius maps or multiple endomorphisms. It's enough to look at rings with a single endomorphism. In fact, the only thing we even need about the site of rings-opposite is that that all objects are compact. So consider the category of rings equipped with an endomorphism. The analogue of (2) is simply the category of pairs (X,f), where X is a sheaf of sets on rings-opposite and f is an endomorphism of X. On the other hand, the analogue of (1) is the category of sheaves Y on [rings with endomorphism]-opposite. These are different categories, and the reason is ultimately that the set N of natural numbers is not compact. Any sheaf can be covered by a family of objects of the site. In (1), this means that every such Y can be covered by a family of [affine schemes with an endomorphism]. Because affine schemes are compact, all orbits of the endomorphism on Y must be compact. But this is not true of all objects in (2). For instance, take the free object on one generator: X=N where f is the translation map. This cannot be covered equivariantly by a family of affine schemes each of which has an endomorphism. But it's pretty easy to fix this by using the category of ind-affine schemes. For any affine scheme T, the scheme NxT=[infinite disjoint union of copies of T] is ind-affine. So we can easily cover any object (X,f) of (2) by a family of ind-affine schemes (i.e. opposite of pro-rings) each of which has an endomorphism: we simply cover X by a family of T's, and then the family of NxT 's covers X equivariantly. That's all that's going on. In the actual lambda world, where we have an action of the monoid M of strictly positive integers which lift Frobenius maps in the appropriate way, the Witt vector functor W^*(T) plays the role of MxT. (And if T is a scheme over the rational numbers, then W^*(T) is literally MxT.) So the W^*(T) won't be coverable by Specs of lambda-rings (unless T is empty). As above, I have no doubt that one could also define lambda-structures on pro-rings, and then take sheaves on the opposite category of them. This is also the reason why much of what I've written is about finite-length Witt functors W_n. At the level of rings, it's usually better not to pass to the limit W_infinity. Because the Spec functor does not send limits to colimits, it's better to take the colimit after applying Spec. (It's also better to take the colimit in the ambient topos rather than in the category of schemes.) Essentially the same fact is that the monad W on affine schemes induced by the comonad W on rings does not agree with the monad W^* on the category of sheaves. If however, we think of the comonad W on rings as a *filtered* comonad W_dot, then it's OK to say that the induced filtered comonad on the category of affine schemes does agree with the filtered comonad W^*_dot. For these reasons, I usually prefer to do all the work with W_n on the category of rings, and then pass to W_infinity and schemes/sheaves/etc at the last possible minute with one wave of the wand of category/topos theory.

   Hi everyone,
   
   I am delighted to see the discussion here and thought I'd jump in and try to help to clear up a few things.
   
   One question was about the difference between (1) sheaves on lambda-rings-opposite and (2) lambda-structures on sheaves on rings-opposite (in the etale topology, say, but that's not too important). The two are different but only for essentially technical reasons, and with a bit of tweaking, they can be made the same. Let me explain.
   
   First, lambda-rings and Witt vectors are all about commuting lifts of Frobenius maps on rings (commutative), but this particular issue has nothing to do with Frobenius maps or multiple endomorphisms. It's enough to look at rings with a single endomorphism. In fact, the only thing we even need about the site of rings-opposite is that that all objects are compact.
   
   So consider the category of rings equipped with an endomorphism. The analogue of (2) is simply the category of pairs (X,f), where X is a sheaf of sets on rings-opposite and f is an endomorphism of X. On the other hand, the analogue of (1) is the category of sheaves Y on [rings with endomorphism]-opposite. These are different categories, and the reason is ultimately that the set N of natural numbers is not compact. Any sheaf can be covered by a family of objects of the site. In (1), this means that every such Y can be covered by a family of [affine schemes with an endomorphism]. Because affine schemes are compact, all orbits of the endomorphism on Y must be compact. But this is not true of all objects in (2). For instance, take the free object on one generator: X=N where f is the translation map. This cannot be covered equivariantly by a family of affine schemes each of which has an endomorphism.
   
   But it's pretty easy to fix this by using the category of ind-affine schemes. For any affine scheme T, the scheme NxT=[infinite disjoint union of copies of T] is ind-affine. So we can easily cover any object (X,f) of (2) by a family of ind-affine schemes (i.e. opposite of pro-rings) each of which has an endomorphism: we simply cover X by a family of T's, and then the family of NxT 's covers X equivariantly.
   
   That's all that's going on. In the actual lambda world, where we have an action of the monoid M of strictly positive integers which lift Frobenius maps in the appropriate way, the Witt vector functor W^*(T) plays the role of MxT. (And if T is a scheme over the rational numbers, then W^*(T) is literally MxT.) So the W^*(T) won't be coverable by Specs of lambda-rings (unless T is empty). As above, I have no doubt that one could also define lambda-structures on pro-rings, and then take sheaves on the opposite category of them.
   
   This is also the reason why much of what I've written is about finite-length Witt functors W_n. At the level of rings, it's usually better not to pass to the limit W_infinity. Because the Spec functor does not send limits to colimits, it's better to take the colimit after applying Spec. (It's also better to take the colimit in the ambient topos rather than in the category of schemes.) Essentially the same fact is that the monad W on affine schemes induced by the comonad W on rings does not agree with the monad W^* on the category of sheaves. If however, we think of the comonad W on rings as a *filtered* comonad W_dot, then it's OK to say that the induced filtered comonad on the category of affine schemes does agree with the filtered comonad W^*_dot.
   
   For these reasons, I usually prefer to do all the work with W_n on the category of rings, and then pass to W_infinity and schemes/sheaves/etc at the last possible minute with one wave of the wand of category/topos theory.
10. • CommentRowNumber10.
    • CommentAuthorJames Borger
    • CommentTimeJul 27th 2014
    • PermaLink
    Author: James Borger
    Format: TextRegarding David's questions about my throwing cold water on F1, I'm actually quite optimistic about applications of lambda/Witt things to the bridge between function fields and number fields. I didn't mean to sound like such a killjoy! My only point is that many of the dreams around F1 are pretty naive, and some of Urs's seemed to be bordering on that. (But I would love to be proved wrong!) In some sense, F1 has become the name attached to any post-Rosetta Stone dreams relating number fields to geometry. (It is a perfectly fine name except that the words "field", "one", and "element" draw attention to the wrong things!) While this is not necessarily so bad and I've even found a number of them helpful in my work on lambda/Witt things, I've also found it best to allow myself very, very, very broad interpretations of these dreams. In my experience, even in successful cases the first ten interpretations of an F1 dream can be some combination of nonsensical, tautologous, and false. Lambda/Witt things are very subtle objects related to global algebraic number theory, and in my experience simple category/topos-theoretic constructions are not so helpful without careful attention to the actual number theory. But I've also found that honest number-theoretic toil, the tight formalism of category theory, and F1 dreams can give a helpful mix of ideas and constraints.

    Regarding David's questions about my throwing cold water on F1, I'm actually quite optimistic about applications of lambda/Witt things to the bridge between function fields and number fields. I didn't mean to sound like such a killjoy! My only point is that many of the dreams around F1 are pretty naive, and some of Urs's seemed to be bordering on that. (But I would love to be proved wrong!) In some sense, F1 has become the name attached to any post-Rosetta Stone dreams relating number fields to geometry. (It is a perfectly fine name except that the words "field", "one", and "element" draw attention to the wrong things!) While this is not necessarily so bad and I've even found a number of them helpful in my work on lambda/Witt things, I've also found it best to allow myself very, very, very broad interpretations of these dreams. In my experience, even in successful cases the first ten interpretations of an F1 dream can be some combination of nonsensical, tautologous, and false. Lambda/Witt things are very subtle objects related to global algebraic number theory, and in my experience simple category/topos-theoretic constructions are not so helpful without careful attention to the actual number theory. But I've also found that honest number-theoretic toil, the tight formalism of category theory, and F1 dreams can give a helpful mix of ideas and constraints.
11. • CommentRowNumber11.
    • CommentAuthorspitters
    • CommentTimeJul 27th 2014
    • PermaLink
    Author: spitters
    Format: MarkdownItexThanks 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]] ?

    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 ?

12. • CommentRowNumber12.
    • CommentAuthorJames Borger
    • CommentTimeJul 27th 2014
    • PermaLink
    Author: James Borger
    Format: TextUrs, regarding you comments in #8, In my opinion, the goal should not be to find an F1 geometry. To the extent that one exists, it is probably something much subtler than what can be measured by formal tests conceived of beforehand. (For instance, the F1 geometry of the point should provide us with a proof of the Riemann hypothesis, and as important as tight formalism is, I think most people would agree that missing ingredient of the Riemann hypothesis is not that our formalism is not good enough.) I think the goal should be to prove new theorems and introduce useful ideas in arithmetic algebraic geometry, inspired by the dreams and analogies of F1 as long as they suggest directions with arithmetic content. That said, I would enjoying learning whether my lambda/Witt picture fits in your cohesive/fracture formalism. If you could tell me the key properties the lambda/Witt picture would need to satisfy, I could have a go at trying to work out whether it does.

    Urs, regarding you comments in #8, In my opinion, the goal should not be to find an F1 geometry. To the extent that one exists, it is probably something much subtler than what can be measured by formal tests conceived of beforehand. (For instance, the F1 geometry of the point should provide us with a proof of the Riemann hypothesis, and as important as tight formalism is, I think most people would agree that missing ingredient of the Riemann hypothesis is not that our formalism is not good enough.) I think the goal should be to prove new theorems and introduce useful ideas in arithmetic algebraic geometry, inspired by the dreams and analogies of F1 as long as they suggest directions with arithmetic content.
    
    That said, I would enjoying learning whether my lambda/Witt picture fits in your cohesive/fracture formalism. If you could tell me the key properties the lambda/Witt picture would need to satisfy, I could have a go at trying to work out whether it does.
13. • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeJul 27th 2014
    • PermaLink
    Author: Urs
    Format: MarkdownItexDear 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.

    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.

14. • CommentRowNumber14.
    • CommentAuthorspitters
    • CommentTimeJul 30th 2014
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    Author: spitters
    Format: MarkdownItexThere 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.

    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.

15. • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeAug 10th 2014
    • (edited Aug 10th 2014)
    • PermaLink
    Author: Urs
    Format: MarkdownItexThanks, 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.

    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.

16. • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeAug 10th 2014
    • (edited Aug 10th 2014)
    • PermaLink
    Author: Urs
    Format: MarkdownItexI 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 * _[Motivation](http://ncatlab.org/nlab/show/Borger%27s+absolute+geometry#Motivation)_. 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.

    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

    • Motivation.

    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.

17. • CommentRowNumber17.
    • CommentAuthorDavid_Corfield
    • CommentTimeMay 29th 2015
    • (edited May 29th 2015)
    • PermaLink
    Author: David_Corfield
    Format: MarkdownItexFor 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.

    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.

18. • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2015
    • PermaLink
    Author: Urs
    Format: MarkdownItexIf 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.

    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.