Question

When we study sheaves of sets (on a space X or a site C) we are often interested in the stalks of the sheaf (at either a point $p:1\to X$ or a left exact, cover-preserving functor $a:C\to Sets$). I wonder how one generalizes this notion to stacks?

Since we can always pass from a stack to an equivalent sheaf, one option might be to say the stalk of a stack is (any category equivalent to) the stalk of the associated sheaf of categories. This seems a little roundabout, though? Is there a more intrinsic definition of these categories?

Answer 1

Here is the categorical way to think about the stalk:

Let $X$ be space, and $x \in X$ a point. Regard $x$ as a map $$x:pt \to X$$.

Then $x$ induces a geometric morphism $$x_*:Sh(pt) \to Sh(X)$$ $$Sh(pt) \stackrel{}{\longleftarrow} Sh(X):x^*$$

(so $x^*$ is left-adjoint to $x_*$ and left-exact).

Now notice that $Sh(pt) \cong Set$. Under this identification, the stalk of $F \in Sh(X)$ at $x$ is the set $x^*\left(F\right)$.

This can be done in the $2$-categorical setting with stacks with no adjustment:

$x$ induces an adjoint pair of $2$-functors

$$x_*:St(pt) \to St(X)$$ $$St(pt) \stackrel{}{\longleftarrow} St(X):x^*$$

and under the identification $St(pt) \cong Gpd,$ the stalk of $\mathscr{X} \in St(X)$ at $x$ is the groupoid $x^*\left(\mathscr{X}\right)$.

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A way of viewing this in a geometric way is by using the etale-realization of the stack $\mathscr{X}$, (explained in arxiv.org/abs/1011.6070 ) which is a topological stack $$L\left(\mathscr{X}\right)$$ equipped with a local homoemosphism to $X$. The stalk can then be viewed as the fiber of this map over $x$.