Comments on: F_un with Manin

By: lieven

lieven — Wed, 17 Sep 2008 15:44:07 +0000

Javier, next time one of your comments doesnt appear at once send me an email and ill fish it out of the Akismet spam-filter...

Fun seminar : well, lets wait for one more week. next week courses start here and apart from the 240hrs i already have to teach anyway this semester (yeah, good news being my 2nd semester is semi-free) i also run a master-course 'seminar noncommutative geometry'. our regular antwerp-seminar starts mid october and i promised to give some talks (combining them with the seminar-course). depending on whether or not students show up for the seminar-course (i rate as highly unlikely, but if they do ill talk about less virtual stuff) i can do some talks on Fun (btw. Raf is also interested in this for entirely different reasons...).

So, maybe we can use this ending-blog as a vehicle to have some interaction between the two seminars?

By: javier

javier — Wed, 17 Sep 2008 14:58:35 +0000

@leon: Thanks for the reference! One more paper to read on...

@lieven: Some days ago I wrote another comment on the notion of coalgebra over F1 that got swallowed. I think your spam filter hates me... Talking different things, I am planning an informal study seminar around F1 here at MPI, starting at the end of the month. If you are interested on participating and sharing your thoughts about it, let me know! Bonn is pretty close to Antwerp, so I am sure we can easily arrange something.

By: leon

leon — Mon, 15 Sep 2008 16:33:54 +0000

Looks like Connes and Consani have also a new paper out on Geometry over F_1, in fact posted just one day before Manin’s http://www.alainconnes.org/docs/ak.pdf

By: javier

javier — Fri, 12 Sep 2008 11:31:55 +0000

Yeah, that seems right... what I an not convinced on is the existence of maps landing on [tex]F1[/tex], not as normal set-theoretical maps anyway (since the field with one element is not really defined as a set). Sort of the same way as you cannot explicitly take a ring map from [tex]F1[/tex] to [tex]\mathbb{Z}[/tex] that would make the integers into a [tex]F_1[/tex] algebra. In SoulÃ©'s paper, the existence of such hypothetical map is 'assumed' to build the notion of scheme, but there is no attempt of providing a realization of it.

My impression was that one should always avoid hitting [tex]F1[/tex] directly, and always try to use things defined over [tex]F1[/tex] as workarounds, that is why I thouhgt of defining algebras and coalgebras as functors on the corresponding categories of vector spaces.

But it is very likely that I got the wrong intuition here, so nevermind...

By: javier

javier — Fri, 12 Sep 2008 08:21:37 +0000

I posted a comment yesterday evening digging deeper on the categorical approach that I hinted above, but since it was tex-intensive, it possibly got lost in the moderation queue. I am not convinced of deducing grouplikeness just from the counit, if you are not allowed to work with linear combinations (because you have no addition).

By: lieven

lieven — Fri, 12 Sep 2008 06:32:28 +0000

@javier : no i was just using definitions. if there is only 1 in the field then this determines the counit and then from coalgebra-defns one gets that all elements must be grouplike and so the coalgebra is cocommutative.
of course i know the weak spot in my ‘argument’. the interpretation of coalgebras as noncommutative geometry is based on the Kostant duality between algebras and coalgebras and this duality breaks down over F-un…
also, i think to understand what Manin has in mind when he calls for a noncomm geometry over F-un. probably he means the gadget bit (the algebra A_X) in Soule’s framework.

By: javier

javier — Thu, 11 Sep 2008 22:15:58 +0000

I've kept thinking about this, and convinced myself about what I said above. Reasoning goes as follows: over an arbitrary field [tex]k[/tex], an algebra [tex]A[/tex] defines a monad [tex]A\bullet[/tex] in the category of [tex]k[/tex]-vector spaces. Conversely, given any monad [tex]T[/tex] on the category of [tex]k[/tex]-vector spaces gives rise to an ordinary algebra [tex]A=T(k)[/tex] by evaluation on the unit object of the category. Thus algebras over a field are the same thing as monads on vector spaces over the same field. Since we have a pretty well defined category of vector spaces over [tex]F_1[/tex], which is the category of sets, an [tex]F_1[/tex]-algebra should be a monad on the category of sets. Commutativity of an algebra can be given in monadic terms by means of a distributive law (which is the fanshy-wanshi way of calling a twisting map) between two copies of the monad, but this is by no means necessary.

Same argument goes for coalgebras and comonads.

By: Kea

Kea — Thu, 11 Sep 2008 01:47:25 +0000

Hmmmm. Ross Street likes to say that quantum sets are coalgebras, and Hopf algebras come along with this philosophy. Co-commutative but non commutative algebras already greatly generalise ordinary ‘commutative’ groups (as sets with only multiplication). This would tie in with the ‘GL(n) is a braid group’ idea of F1 maths. Must check out the paper … now that the LHC news has died down.

By: javier

javier — Wed, 10 Sep 2008 19:03:26 +0000

I assume your reassoning relies on the interpretation of groups as Hopf algebras over [tex]F_1[/tex], right? If this is the case, same argument would allow to state that all algebras over the field with one element are division rings, and I don't think that is what these people have in mind.

On the other hand, if you think about "going to [tex]F_1[/tex]" as "forgetting addition", an algebra should be something like a monoid (or a monad), and the right dual would be a comonad.

Still I am myself not sure this is the right approach...