Schreiber higher group characters

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

Funtorial reformulation

For $k \in \mathbb{N}$, $k \geq 1$, let $\Pi_1^k(S^1)$ 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 $\Pi_1^1(S^1) = \mathbf{B}\mathbb{Z}$.

Then

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

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

In particular the projection map

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

is precisely the corresponding weak Cayley table.

Remarks

• For $k=1$ we have that

  $$Funct(\Pi_1^1(S^1), \mathbf{B}G) = Funct(\mathbf{B}\mathcal{Z}, \mathbf{B}G) = \Lambda G$$

  is the “loop groupoid” of $G$ as discussed in

• Simon Willerton , …,

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