Stalks as colimits

Question

I'm currently trying to understand how we can view stalks of sheaves as the colimits of the sections i.e. $$\mathscr{F}_x = \varinjlim_{x \in U} \mathscr{F}(U)$$ but I have some issues with my understanding of colimits.

The way I've learned about these is that we usually take the colimit of a diagram $$F: I \to \mathcal{C}$$ is an object which satisfies some universal property etc.

So coming back to this case with sheaves and stalks I'm wondering what are $I$ and $\mathcal{C}$ supposed to be here?

Answer 1

Let $\mathcal{C}$ be an arbitrary category and $I$ a small category. Now let $F: I \longrightarrow \mathcal{C}$ be any functor, a colimit of this functor is at first an object of $\mathcal{C}$, written $\mathrm{colim}_{i \in I}F(i)$, together with morphisms ($f:i \longrightarrow j$ in $I$) $\require{AMScd}$ \begin{CD} F(i) @>{F(f)}>> F(i)\\ @VVV @VVV\\ \mathrm{colim}_{i \in I}F(i) @>{=}>> \mathrm{colim}_{i \in I}F(i) \end{CD} and we require these datum to be universal. That being said, the diagram above is an "intial object" among diagrams of the form $\require{AMScd}$ \begin{CD} F(i) @>{F(f)}>> F(i)\\ @VVV @VVV\\ C @>{=}>> C \end{CD} From those above, you see that $F(f)$ are morphisms in $\mathcal{C}$ and $F(i)$ are objects in $\mathcal{C}$. In case colimits exist for all $F$ indexed by $I$, we say that $\mathcal{C}$ is cocomplete. And colimit becomes a functor $$\mathrm{colim}_{i \in I}: \mathrm{Func}(I,\mathcal{C}) \longrightarrow \mathcal{C}.$$ The universal property of colimit implies that it is left adjoint to the diagonal functor $\Delta: \mathcal{C} \longrightarrow \mathrm{Func}(I,\mathcal{C})$ sending $C \longmapsto (i \longmapsto C).$ (In other words, $\Delta$ send each object to a constant sequence indexed by $I$) $$\mathrm{Hom}_{\mathcal{C}}(\mathrm{colim}_{i \in I}F(i), C) = \mathrm{Hom}_{\mathrm{Func}(I,\mathcal{C})}(F,\Delta(C)).$$

Now let me come back to your case:

• $I$ is the category of open subsets containing $x$. It is a category whose objects are open subsets and for every two such $U,V$ you have $$\mathrm{Hom}(U,V) = \begin{cases} \bullet & x \in U \subset V \\ \varnothing & \mathrm{otherwise} \end{cases}.$$ This is similar to when you define a presheaf $F$ on a topological space $X$, instead of writing all datum out. You can simply announce it to be a functor $$F: \mathrm{Open}(X)^{op} \longrightarrow \text{some category you want}$$ where $\mathrm{Open}(X)$ has objects as open subset of $X$ and morphisms defined analogously to the above except that you do not require them to contain $x$.
• $\mathcal{C}$ can be whatever category you want to work with.
• Stalk is a good example of the so-called filtered colimit. It is filtered because we can take intersections and hence is convenient for definining operations. For instance, when $\mathcal{C} = \mathbf{Ab}$, the category of abelian groups, stalks themselves must be abelian groups: the elements of $\mathscr{F}_x$ are pairs $[f,U]$ with $f \in \mathscr{F}(U)$ where $[f,U] = [g,V]$ if $f_{\mid U \cap V} = g_{\mid U \cap V}$ and we define the addition by $$[f,U] + [g, V] = [f_{\mid U \cap V} + g_{\mid U \cap V}, U \cap V],$$ and without being filtered, there are no canonical way to define additions. Most examples showing up in practice are filtered.

For a presheaf $F: \mathrm{Open}(X)^{op} \longrightarrow \mathcal{C}$, we say:

• $F(U)$ is the set of sections over $U$.
• $F(V) \longrightarrow F(U)$ is the restriction of sections on $V$ to $U$ whenever $U \subset V$.

In fact, these terminologies are not used in an absolutely correct way as you may realize that you have nothing to talk about restrictions. But as I pointed out, it is a fake good style to remind a novice of the most typical case: sheaf of continuous functions on $X$.

• $F(U) = \left \{f: U \longrightarrow \mathbb{R} \ \text{continuous}\right \}.$

• $F(V) \longrightarrow F(U), f \longmapsto f_{\mid U}$.

In this case, restrictions are really restrictions. Another motivation for these terminologies may come from the original definition of a sheaf: a sheaf is a sheaf of an étale space. An étale space over $M$ is simply a local homeomorphism $p:E \longrightarrow M$. Then you have a sheaf of sections

• $F(U) = \left \{s: U \longrightarrow E \ \text{continuous} \mid f \circ s = \mathrm{id}_U \right \}$.
• $F(V) \longrightarrow F(U), s \longmapsto s_{\mid U}$ because $f \circ s = \mathrm{id}_V$ then $f \circ s_{\mid U} = \mathrm{id}_U$.

If moreover, you are asking that why use the word "section"? Then the example of étale space already provides the answer. Here I took the picture on wikipedia on vector bundles (intuitively, they are same things).

You see that $E \longrightarrow M$ is a kind of étale space. You collapse every vertical line to its middle point, resulting in $M$; or you can say, you project every point on a vertical line to a point. The preimage of any curve interval $U$ on $M$ is a strip that $U$ cuts along the middle. A section over a curvy interval $U$ on $M$ is simply a way to going back from $U$ to $E$ (because if you project on $U$ again, it is identity) or equivalently speaking, a way of cutting along the strip (the preimage) of $U$, but not necessarily being the middle cut.

Slogan. A section is a way of cutting the total space and it could be different from the original cut (the original cut is often called the zero section in case every vertical line is a line e.g. $\mathbb{R},\mathbb{C},...$)

The stalk of the sheaf of sections then must be the fiber of the étale space.