For $G$ an abelian group, the Eilenberg-Maclane space $K(G, n)$ represents singular cohomology $H^n(-; G)$ with coefficients in $G$ on the homotopy category of CW-complexes. If $n > 1$, then $G$ must be abelian, but for $n = 1$ there are also Eilenberg-Maclane spaces for $G$ nonabelian, so one might imagine that they represent some kind of nonabelian cohomology $H^1(-; G)$. Is this functor known by a better name, and what's known about it? In particular,
- Does it have an alternate definition along the lines of the usual definition of singular cohomology?
- Is there a universal coefficient theorem for it?