We have seen before that the symmetric group S(n) has a F-un interpretation as the linear group GL(n) over the field with one element. Hence, we can ask whether there is also a F-un interpretation of the n-string braid group and of the above group-morphism.
Over F_un vectorspaces are pointed sets. One can also do linear algebra over the F_un extension fields F_un^n. Vectorspaces are then pointed sets having a free action of n-th roots of unity.
The followers of F-un have their own collection of semi-secret texts, surrounded by whispers, of which they try to decode every single line in search of enlightenment. Fortunately, you do not have to search the shelves of the Bibliotheque National in Paris, but the depths of the internet to find them as huge, bandwidth-unfriendly, scanned documents.
In recent years several people spend a lot of energy looking for properties of an elusive object : the field with one element or in French : “F-un”. One of the motivations is the hope to prove the Riemann hypothesis by mimicking Weil’s proof in the case of curves.
Yuri I. Manin : Cyclotomy and analytic geometry over F_un, arXived on September 9th, 2008 arXiv:0809.1564v1
Alain Connes and Caterina Consani : On the notion of geometry over F_un, arXived on September 17th,200b as arXiv:0809.2926
Alain Connes, Caterina Consani and Matilde Marcolli : Fun with F_un, arXived on June 14th, 2000 as arXiv:0806.2401
Nikolai Durov : New Approach to Arakelov Geometry, arXived April 16th, 2007 arXiv:0704.2030v1
Yuri I. Manin : The notion of dimension in geometry and algebra, Bull. AMS 43 (2006) 139-161.
Anton Deitmar : F_un schemes and toric varieties, arXived on August 7th,2006 as arXiv:math.NT/0608179