# Schreiber higher group characters

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

# Funtorial reformulation

For $k\in ℕ$, $k\ge 1$, let ${\Pi }_{1}^{k}\left({S}^{1}\right)$ 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 ${\Pi }_{1}^{1}\left({S}^{1}\right)=Bℤ$.

Then

Proposition. The $k$-conjugacy classes of $G$ are precisely the isomorphism classes of the groupoid

$\mathrm{Funct}\left({\Pi }_{1}^{k}\left({S}^{1}\right),BG\right)\phantom{\rule{thinmathspace}{0ex}}.$Funct(\Pi_1^k(S^1), \mathbf{B}G) \,.

In particular the projection map

$\mathrm{Funct}\left({\Pi }_{1}^{k}\left({S}^{1}\right),BG\right)\to >\mathrm{Funct}\left({\Pi }_{1}^{k}\left({S}^{1}\right),BG\right){/}_{\sim }$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

$\mathrm{Funct}\left({\Pi }_{1}^{1}\left({S}^{1}\right),BG\right)=\mathrm{Funct}\left(B𝒵,BG\right)=\Lambda G$Funct(\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)