higher group characters

There is a functorial reformulation of the theory of higher group characters.

Funtorial reformulation

For kk \in \mathbb{N}, k1k \geq 1, let Π 1 k(S 1)\Pi_1^k(S^1) be the groupoid with kk objects which is weakly equivalent with respect to the folk model structure to the fundamental groupoid of the circle S 1S^1.

In particular Π 1 1(S 1)=B\Pi_1^1(S^1) = \mathbf{B}\mathbb{Z}.


Proposition. The kk-conjugacy classes of GG are precisely the isomorphism classes of the groupoid

Funct(Π 1 k(S 1),BG). Funct(\Pi_1^k(S^1), \mathbf{B}G) \,.

In particular the projection map

Funct(Π 1 k(S 1),BG)>Funct(Π 1 k(S 1),BG)/ Funct(\Pi_1^k(S^1), \mathbf{B}G) \to \gt Funct(\Pi_1^k(S^1), \mathbf{B}G)/_\sim

is precisely the corresponding weak Cayley table.


  • For k=1k=1 we have that

    Funct(Π 1 1(S 1),BG)=Funct(B𝒵,BG)=ΛG Funct(\Pi_1^1(S^1), \mathbf{B}G) = Funct(\mathbf{B}\mathcal{Z}, \mathbf{B}G) = \Lambda G

    is the “loop groupoid” of GG as discussed in

  • Simon Willerton , …,

i.e. the action groupoid of GG acting on itself by conjugation.

Created on December 10, 2008 at 10:46:56. See the history of this page for a list of all contributions to it.