Motive Madness II

Recall that Kapranov and Smirnov have also been thinking about the field with one element. They say an affine line over $F_1$ should be zero along with all the roots of unity.

Looking at polynomials with $F_1$ coefficients, the group $GL(n,F_1[x\left]\right)$ is just the braid group on $n$ strands. For example, $3\times 3$ matrices are associated with the three strand braid group, as often discussed. Then the map $B_n\to S_n$ is thought of as the $q\to 1$ limit, since the symmetric group acts on sets as vector spaces.

The field $F_1\left(n\right)$ extends $F_1$ by containing zero and the set of all nth roots of unity. A vector space over this field is a pointed set (marked by zero) with an action by the roots of unity. Direct sum and smash product become the operations on such spaces.

Note that Weber and others have considered the category of pointed sets as a 2-categorical Cat analogue of subobject classifier, and the category Set plays the role here of a one element set. There seem to be a number of ways in which the field $F_1$ introduces new topos theoretic arrows.