# Canonical group structure of germs

Given a topological space $X$ and a presheaf of abelian groups on $X$, $A$, we can construct the set of germs at a point $x\in X$ by taking $\mathscr{A}_x=\lim\limits_{\rightarrow} A(U)$ where $x\in U\subseteq X$ and $U$ is open. In Sheaf Theory by Bredon, he claims there is a canonical group structure on $\mathscr{A}_x$.

What is the canonical group structure of the direct limit?

I would think if $[s]_x, [t]_x$ are germs then $[s]_x+[t]_x=[s+t]_x$ but, if $s,t$ are not in the same group, would we first need to restrict their sections until they are and then add them?

• Right. It might be good to have notation that explicitly keeps track of $U$. Show that none of this depends on choices. – Hoot May 25 '15 at 17:25
• @Hoot, that's what I was worried about haha. – Eoin May 25 '15 at 17:26

As Hoot suggested, the group structure is actually $[(s,U)]_x+[(t,V)]_x=[(s|_{U\cap V}+t|_{U\cap V}, U\cap V)]_x$