nLab
Baer sum

Idea

The Baer sum is the natural addition operation on abelian group extensions.

For $G$ a group and $A$ an abelian group, the extensions of $G$ by $A$ are classified by the degree-2 group cohomology

$$H_{\mathrm{Grp}}^2(G,A)=H^2(BG,A)=H(BG,B^{2}A).$$H^2_{Grp}(G,A) = H^2(\mathbf{B}G, A) = H(\mathbf{B}G, \mathbf{B}^2 A) \,.

On cocycles $BG\to B^{2}A$ there is a canonical addition operation coming from the additive structure of $A$, and the Baer sum is the corresponding operation on the extensions that these cocycles classify.

Definition

Below are discussed several different equivalent ways to define the Baer sum

On concrete cocycles

A cocycle in degree-2 group cohomology $H_{\mathrm{Grp}}^2(G,A)$ is a function

$$c:G\times G\to A$$c : G \times G \to A

satisfying the cocycle property.

Definition

Given two coycles $c_1,c_2:G\times G\to A$ their sum is the composite

$$(c_1+c_2):G\times G\stackrel{{\Delta }_{G\times G}}{\to }(G\times G)\times (G\times G)\stackrel{(c_1,c_2)}{\to }A\times A\stackrel{+}{\to }A$$(c_1 + c_2) : G \times G \stackrel{\Delta_{G \times G}}{\to} (G \times G) \times (G \times G) \stackrel{(c_1,c_2)}{\to} A \times A \stackrel{+}{\to} A

of

• the diagonal on $G\times G$;

• the direct product $(f,g)$;

• the group operation $+:A\times A\to A$.

Hence for all $g_1,g_2\in G$ this sum is the function that sends

$$(c_1+c_2):(g_1,g_2)\mapsto c_1(g_1,g_2)+c_2(g_1,g_2)$$(c_1 + c_2) : (g_1, g_2) \mapsto c_1(g_1,g_2) + c_2(g_1, g_2)

On abstract cocycles

As discussed at group cohomology, a cocycle $c:G\times G\to A$ is equivalently a morphism of 2-groupoids from the delooping groupoid $BG$ of $G$ to the double-delooping 2-groupoid $B^{2}A$ of $A$:

$$c_1,c_2:BG\to B^{2}A.$$c_1,c_2 : \mathbf{B}G \to \mathbf{B}^2 A \,.

Since $A$ is an abelian group, $matbf{B}^{2}A$ is naturally an abelian 3-group, equipped with a group operation $+:(B^{2}A)\times (B^A)\to B^{2}A$.

With respect to this the sum operation is

$$c_1+c_2:BG\stackrel{{\Delta }_{BG}}{\to }BG\times BG\stackrel{(c_1,c_2)}{\to }{B}^{2}A\times B^{2}A\stackrel{+}{\to }{B}^{2}A$$c_1 + c_2 : \mathbf{B}G \stackrel{\Delta_{\mathbf{B}G}}{\to} \mathbf{B}G \times \mathbf{B}G \stackrel{(c_1,c_2)}{\to} \mathbf{B}^2 A \times \mathbf{B}^2 A \stackrel{+}{\to} \mathbf{B}^2 A

On short exact sequences

For $0\to A\to {\hat{G}}_i\to G\to 0$ for $i=1,2$ two short exact sequences of abelian groups, their Baer sum is

$${\hat{G}}_1+{\hat{G}}_2≔{+}_{*}{\Delta }^{*}{\hat{G}}_1\times {\hat{G}}_2$$\hat G_1 + \hat G_2 \coloneqq +_* \Delta^* \hat G_1 \times \hat G_2

The first step forms the pullback of the short exact sequence along rhe diagonal on $G$:

$$\begin{array}{ccc}A\oplus A& \to & A\oplus A\\ \downarrow & & \downarrow \\ {\Delta }^{*}({\hat{G}}_1\oplus {\hat{G}}_2)& \to & {\hat{G}}_1\oplus {\hat{G}}_2\\ \downarrow & & \downarrow \\ G& \stackrel{{\Delta }_G}{\to }& G\oplus G\end{array}$$\array{ A \oplus A &\to& A \oplus A \\ \downarrow && \downarrow \\ \Delta^* (\hat G_1 \oplus \hat G_2) &\to& \hat G_1 \oplus \hat G_2 \\ \downarrow && \downarrow \\ G &\stackrel{\Delta_G}{\to}& G\oplus G }

The second forms the pushout along the addition map on $A$:

$$\begin{array}{ccc}A\oplus A& \stackrel{+}{\to }& A\\ \downarrow & & \downarrow \\ {\Delta }^{*}({\hat{G}}_1\oplus {\hat{G}}_2)& \to & {+}_{*}{\Delta }^{*}({\hat{G}}_1\oplus {\hat{G}}_2)\\ \downarrow & & \downarrow \\ G& \to & G\end{array}$$\array{ A \oplus A &\stackrel{+}{\to}& A \\ \downarrow && \downarrow \\ \Delta^* (\hat G_1 \oplus \hat G_2) &\to& +_* \Delta^*(\hat G_1 \oplus \hat G_2) \\ \downarrow && \downarrow \\ G &\to& G }

Related concepts

• cup product

References

Lecture notes include for instance

• Patrick Morandi, Ext Groups and Ext Functors (pdf)