Here is my approach to relate two subcategories of $\textbf{hTop}$ and $\textbf{Gp}$ respectively:
Let $\mathcal{C}$ denote the full subcategory of $\textbf{hToP}$ consisting of acyclic( i.e. reduced homology in all degree is zero ) CW complex and $\mathcal{D}$ denote the category of perfect groups (i.e. commutator is itself). There is functor $\mathcal{F} = \pi_{1} : \mathcal{C} \rightarrow \mathcal{D}$ takes space $X$ to its fundamental group $\pi_{1}(X)$. We have also another functor $\mathcal{G} : \mathcal{D} \rightarrow \mathcal{C}$ given as if $P$ is any perfect group then define $\mathcal{G}$(P) to be the homotopy fiber of the plus construction map $ BP \rightarrow BP^{+}$. The above two functors seems too natural to me. So mine natural questions are the following:
- Are $\mathcal{C}$ and $\mathcal{D}$ equivalent categories?
- Is $\mathcal{F}$ related to $\mathcal{G}$ (I mean whether they are in adjunction, equivalence pair etc.)?
I computed the composition $\mathcal{FoG}$: Let $P$ be a perfect group and $F(f)$ denote the homotopy fiber of $ BP \rightarrow BP^{+}$. Then $\mathcal{FoG}(P) = \pi_{1}(F(f))$ and the counit map $\mathcal{FoG}(P) = \pi_{1}(F(f)) \rightarrow P $ is the universal central extension map which in particular says that if we again repeat the same process turns out to be identity. So this is in a way saying that $\mathcal{FoG}$ may not be identity but its square is identity. I don't know what it means. Any help would be appreciated.
