# Two definitions of central extensions of simplicial groups

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:

1. given $E'$, construct $E$

2. 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.

• In (1) I believe the fiber sequence should only be required to be a homotopy fiber sequence, in order to obtain a homotopy invariant notion. In contrast, Definition (2) seems to be hopelessly non homotopy invariant, and so you may want to forsake it. On the other hand, (1) only requires $A$ to be a double loop space, while (2) requires it to be a simplicial abelian group. If it's the latter type of kernel that you're after you may be interested in an intermediate notion where the kernel is an infinite loop space, see Remark 2.4.3 in arxiv.org/abs/1612.02608. – Yonatan Harpaz Aug 18 '17 at 8:07
• For a homotopy inverse functor, you could set $H':= G\bar{W}H$ and let $E':=H'\times_{\bar{W}A}WA$, where the morphism $H'\to \bar{W}A$ is given by adjunction and $WA$ is the principal $A$-bundle over $\bar{W}A$. – Jon Pridham Aug 18 '17 at 9:14
• Perhaps it would help to use the description of fibrations using twisted cartesian products together with actions. I have a vague recollection of a paper on this from `way-back' possibly in the Duke MJ. – Tim Porter Aug 18 '17 at 15:22
• There is a categorical theory of central extensions for a wide class of categories. I think $sGrp$ is one of them. see D Bourn, M. Gran, Central extensions in semi-abelian categories, J. Pure Appl. Alg. 175 (2002) 31-44 – Tim Porter Aug 18 '17 at 19:03
• Thanks everybody for the comments, I've edited the post. @TimPorter, thanks for the references, as far as I remember simplicial objects in any semi-abelian (or algebraic ?) category again forms semi-abelian category. It will be interesting to see what their definition gives for $\mathrm{sGrp}$. – res Aug 19 '17 at 13:40