I'm trying to better understand stalks of sheaves, but my question here is mostly categorical.
Let $X$ be a topological space, $p \in X$, and $\mathcal{F}$ a sheaf on $X$ (into some concrete category). In defining a stalk, we consider the directed system of open neighborhoods of a point $p$ in a space $X$, with the morphisms being inclusions. Then we apply $\mathcal{F}$ to this directed system. On the wikipedia page on direct limits (https://en.wikipedia.org/wiki/Direct_limit), they give this as an example, and they say that we get an "associated directed system" $(\mathcal{F}(U), r^V_U)$ where $r^V_U$ are the restriction maps.
Here's my question. If we have a directed system $(A_i, f^j_i)$ where $f^j_i:A_i \to A_j$ for $i \le j$, and we apply a contravariant functor $\mathcal{F}$, we get $(\mathcal{F}(A_i), \mathcal{F}(f^j_i))$ where $\mathcal{F}(f^j_i):A_j \to A_i$ for $i \le j$. How is the output a directed system? It looks more like an inversely directed system to me. Do we reverse the ordering on our indexing set or something like that?
You are right ; if $\mathcal{F}$ is a contravariant, then the image of a directed system will be inversely directed (or codirected). If you want the image of your directed system to be a directed system, you need to use a covariant functor.
What happens is that in the case of a topological space $X$ and a sheaf $\mathcal{F}$, the open neighborhoods of $p$ form a directed system under reversed inclusion. So $U\leq V$ means in fact that $V\subset U$! So your sheaf, being a contravariant functor on the category of open subsets of $X$, actually becomes a covariant functor on the directed system considered, which is why $(\mathcal{F}(A_i), \mathcal{F}(f^j_i))$ is really a directed system.