Show that a certain functor preserves colimits and finite limits by verifying it on the stalks of sheaves

Question

In Moerdijk, Classifying spaces and classifying topoi, on page 22, we have a functor from the topos $\mathcal BG$ of right $G$-sets ($G$ is a group) to the topos $Sh(X)$, the sheaves (étale spaces) over a topological space $X$. We want to prove that it preserves colimits and finite limits. The Author says that "it suffices to check this for the stalk at each point $x\in X$". He then computes the stalk $F(S)_x$ for every $S\in \mathcal BG$ (call $F$ the functor) and finds that $F(S)_x\cong S$ for every $x$, so it is clear that $F$ preserves colimits and finite limits.

My question is: why does it suffice to study stalks? My idea is that, as always, the sheaf structure allows one to "recover" the global properties later, and focus on what happens on stalks. Can one help me formalizing this in relationship to the specific context of limits and colimits?

Thank you in advance.

Answer 1

For any functor between any categories, there are canonical maps $\varphi:F(\operatorname{lim}S_i)\rightarrow\operatorname{\lim} F(S_i)$ and $\psi:\operatorname{colim}F(S_i)\rightarrow F(\operatorname{colim}S_i)$. To say that $F$ preserve limits/colimits means that these maps are isomorphisms.

Now, if the target category is a category of sheaves over some topological space, there is a useful criterion for a map to be an isomorphism : it is enough to check it on stalks.

You get by composition $(F(\lim S_i))_x\overset{\varphi}\rightarrow (\lim F(S_i))_x\rightarrow \lim F(S_i)_x$. If this is a finite limit, the last map is an isomorphism (stalks commute with finite limits), hence $\varphi:F(\operatorname{lim}S_i)\rightarrow\operatorname{\lim} F(S_i)$ for any map $(F(\operatorname{lim}S_i))_x\rightarrow\operatorname\lim F(S_i)_x$ is an isomorphism.

The same works for colimits, but in that case, you don't need to assume that the colimit is finite because stalks commute with arbitrary colimits.