Central Extensions of Simplicial Groups and Presheaves of Simplicial Groups

We show that path components of central extensions of a simplicial group $G$ by a simplicial abelian group $A$ are in bijection with the homotopy classes of maps between classifying spaces $[\overline{W}G, \overline{W}^2A]$. Generalizing the classical correspondence between isomorphism classes of central extensions of a group $G$ by an abelian group $A$ and homotopy classes of maps $[\overline{W}G, \overline{W}^2A]$. We then prove an analogous theorem for central extensions of presheaves of simplicial groups. Finally, we use the correspondence to show that the cup product factors through a Heisenberg central extension.