Schreiber
higher group characters

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

Funtorial reformulation

For k, k1, let Π 1 k(S 1) be the groupoid with k objects which is weakly equivalent with respect to the folk model structure to the fundamental groupoid of the circle S 1.

In particular Π 1 1(S 1)=B.

Then

Proposition. The k-conjugacy classes of G 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.

Remarks

  • For k=1 we have that

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

    is the “loop groupoid” of G as discussed in

  • Simon Willerton , …,

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

Created on December 10, 2008 10:46:56 by Urs Schreiber (134.100.221.205)