This is a cross-post from MSE.
Let $\overline W$ be a classifying space functor on $\mathrm{sGrp}$ with $G$ be a corresponding left adjoint (Kan's loop group).
Def 1 : a sequence of maps $A\to E\to H$ in $\mathrm{sGrp}$ is called the central extension (of $H$ by $A$) if $$ \overline W E\to \overline W H\to \overline W^2 A $$
is a fiber sequence in $\mathrm{sSet}$.
Def 2 :a sequence of maps $A\to E\to H$ in $\mathrm{sGrp}$ is called the central extension (of $H$ by $A$) if for any $n$ $$ A_n \to E_n \to H_n $$ is the central extension of groups.
Question : are these two definitions equivalent ?
Certainly, any central extension in sense of (2) produces a fiber sequence $$ \overline W A\to \overline W E\to \overline W H $$ which can be shown to be a $\overline W A$-principal fibration (I believe that principal action should be induced from $A\times E\to E, \ (a,e)\mapsto ae$), hence produces a central extension in sense of definition (1).
But what about the opposite direction ? My only idea is to try to cook up a set of 2-cocycles $c_n\in[K(H_n,1),K(A_n,2)]$ from a classifying map $\overline W H\xrightarrow w \overline W^2 A$. Weak equivalence $\overline W H\simeq d N H$ may help ($N$ is a nerve functor: $(N H)_{n,m}=K(H_n,1)_m$ and $d$ is a diagonal of bisimplicial set), but I have no idea, how to extract $K(A_n,2)$ from $\overline W^2 A$.
EDIT
As Yonatan mention, fiber sequence in definition (1) is in fact homotopy fiber sequence, actually in (1) I think about arbitrary simplicial map $\overline W H\xrightarrow w \overline W^2 A$ with a homotopy fiber (in $\mathrm{sGrp}$) $\overline W E$. In the same manner I thought about equivalence of definitions, i.e. (1) and (2) are equivalent if there exists a (homotopy) commutative diagram
$\require{AMScd}$ \begin{CD} A @>>> E' @>>> H' \\ @| @VVV @VV{\simeq}V \\ A @>>> E @>>> H \end{CD}
Here bottom row is a central extension of simplicial groups in sense of (2) and top row is obtained, as in Jon's comment, by pulling back the universal bundle $A\to WA\to\overline W A$ along the adjoint map $H'=G\overline W H\to \overline W A$. Basically this is the same as saying that $E'$ is a homotopy fiber of $H'\to \overline W A$. So my problem is in fact:
given $E'$, construct $E$
construct a map $E'\to E$ making the diagram above homotopy commutative.
Finally, $A$ is assumed to be a simplicial abelian group, otherwise, as mentioned it should be possible to construct some weird example with $E_2$-space.
