math.CT — Category Theory

The third unit in the adjoint tower of group involutions

Contributed by Zhenyu Sun

The functor $G\mapsto\operatorname{Hom}(C_2,G)$ determines an adjoint tower through successive Eilenberg--Moore comparison functors. We prove that the unit of the third adjunction need not be an isomorphism, answering a question of Bergman. For the generalized dihedral group $H=(C_4\times C_4)\rtimes C_2$, with inversion action, we compute $F_3V_3H\cong H\times C_2$, with counit the first projection. The unit at $V_3H$ is injective, with underlying domain and codomain of cardinalities $20$ and $40$. For an arbitrary group $G$, let $E(G)=F_2V_2G$ and let $K_G$ be the kernel of the second counit. We prove that $K_G$ is central and that $F_3V_3G\cong E(G)/K_G[2]$, where $K_G[2]$ consists of the elements of $K_G$ whose square is one. The third unit at $V_3G$ is an isomorphism precisely when $K_G$ has no element of order four. A central extension of $H$ of order $128$ supplies such an element and yields the stated description of $F_3V_3H$.