Take the 2-minute tour ×  Eilenberg and Mac Lane showed that
given a group $G$ there exists a pointed topological space $X_G$
such that $\pi(X_G,\bullet)\cong G$. It is obviously a way to
"invert direction" to the functor $\pi_1\colon
\mathbf{Top}^\bullet\to \mathbf{Grp}$ to a functor $\mathcal
K\colon \mathbf{Grp}\to \mathbf{Top}^\bullet$ such that
$\pi_1(\mathcal K(G),\bullet)\cong G$ (almost by definition).
This is equivalent to say that there exists a natural
transformation (equivalence, in this case) between
$\pi_1\circ\mathcal K$ and $\mathbf{1}_{\mathbf{Grp}}$, which
turns out to resemble some sort of counity.  It would be
wonderful if I could define an adjunction between the two
categories in exam, given by the two functors. Everytime I try
to think about some sort of unity to this hypotetical adjunction
I poorly fail: considering the vast literature in the field of
algebraic topology, I believe in only two possible cases. The
first, nothing interesting arises from this adjunction. The
second, there i