Here is a quick question about definitions.

Let $\mathcal{F}$ be a sheaf of abelian groups of a space $M$. The stalk $\mathcal{F}_x$ at $x\in M$ is defined by $$\mathcal{F}_x=\{(U,s):x\in U\subseteq M,U\text{ open },s\in\mathcal{F}(U)\}/\sim$$ where $(U,s)\sim(V,t)$ if there exists an open set $W$ such that $x\in W\subseteq U\cap V$ and $s|_W=t|_W$.

Now I am told that a homomorphism of sheaves $\varphi:\mathcal{F}\to\mathcal{G}$ induces a homomorphism of stalks $\varphi_x:\mathcal{F}_x\to\mathcal{G}_x$. However, I was not given any group structure for a stalk.

What is the group structure of the stalk $\mathcal{F}_x$?

My guess is that it is defined as $$[(U,s)]+[(V,t)]=[(U\cap V,s|_{U\cap V}+t|_{U\cap V}].$$ Is that correct?


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