Where does the name "filtered colimit" come from? There are a lot of articles which explain what filtered colimits are (e.g. https://ncatlab.org/nlab/show/filtered+limit), but I couldn't find why they are named "filtered colimits".
It doesn't look like they have something to do with filters (in order theory), so I'm wondering.
Any ideas?
 A: Filtered categories are meant to generalize (upward) directed posets, i.e. nonempty posets $P$ such that for all $p,q \in P$, there is $r \in P$ with $p \leq r \geq q$. In particular, a poset $P$ is directed if and only if it is filtered when regarded as a category. The main point of directed posets (and filtered categories more generally) in category theory is to take colimits indexed by them. The great thing is that in $\mathsf{Set}$ and in categories defined by finitary relations / operations, finite limits commute with filtered colimits. I'll give an explanation for the term "filtered" in a moment.
First I want to emphasize the point I made, in connection with the question in the comments: If $P$ is a nonempty poset, then it's true that the set $\mathrm{Dir}(P)$ of directed subsets of $P$ forms a directed poset ordered under inclusion. But that's not really the point. The point is that if you have a poset $P$ which is itself directed (e.g. $P$ could be an upward-directed subset of another poset $Q$), then $P$ is a filtered category.
So it might look like it's an unfortunate clash of terminology that the generalization of a directed poset is called "filtered" -- what should the generalization of a filter be then? Let me argue that filtered categories are actually more closely related to filters than they might appear.
First, there's a shift in perspective that has happened here. Most of the time in logic, you don't talk about an abstract poset being directed. Rather, you talk about directed subsets of other posets. Similarly, you talk about filters on posets. This shift in perspective is actually mirrored in category theory. Namely, instead of taking an ordinary colimit of a functor $F: P \to \mathcal{C}$, you can take a weighted colimit. The data for this is an indexing category $P$, a functor $\phi: P^\mathrm{op} \to \mathsf{Set}$, and a functor $F: P \to \mathcal{C}$; the colimit of $F$ weighted by $\phi$ is an object $\phi \ast F$ of $\mathcal{C}$ with a certain universal property you can read about on the nlab page. The ordinary colimit of $F$ is the colimit of $F$ weighted by the presheaf which is constant at the one-elment set. It turns out that $\phi \ast F$ is also the ordinary colimit of a certain functor $\mathsf{el}(\phi) \to \mathcal{C}$, where $\mathsf{el}(\phi)$ is the category of elements of $\phi$. It's a somewhat nontrivial fact that finite limits commute with $\phi$-weighted colimits in $\mathsf{Set}$ iff they commute with ordinary $\mathsf{el}(\phi)$-colimits in $\mathsf{Set}$.
When $P$ is a poset and $\phi$ takes values in subsingleton sets, then the data of $\phi$ is equivalent to specifying a downward-closed subset $S_\phi \subseteq P$ ($S_\phi$ is the set of $p \in P$ such that $\phi(p)$ is nonempty). The category of elements $\mathsf{el}(\phi)$ is in fact just $S_\phi$ with the induced ordering from $P$. So the $\phi$-weighted colimit of $F$ is just the colimit of $F$ restricted to $S_\phi$. And $\phi$-weighted colimits commute with finite limits in $\mathsf{Set}$ iff $S_\phi$ is an (upward) directed poset, i.e. iff $S_\phi^\mathrm{op}$ is a filter in $P^\mathrm{op}$.
To sum up: if you have a downward-closed set $S$ in a poset $P$ and a functor $F: P \to \mathcal{C}$, you can take the colimit of $F$ weighted by S, which is just the colimit of $F$ restricted to $S$. Such colimits commute with finite limits in $\mathsf{Set}$ if and only if $S^\mathrm{op}$ is a filter in $P^\mathrm{op}$. So from a categorical perspective, the key fact about filters is that colimits weighted by them (or rather, their opposites) commute with finite limits in $\mathsf{Set}$, and this is the key property generalized to filtered colimits in the case when the colimit is an ordinary colimit (presheaves $\phi$ such that $\phi$-weighted colimits commute with finite limits in $\mathsf{Set}$ are usually called flat). The only weird thing is that opposite categories have to be taken at some point to keep the terminology consistent.
A: A short answer about etymology: there is a general notion of filter for posets which coincides with the usual one as soon as the poset is a meet-semilattice: a filter is a down-directed upwards-closed set.
Moreover a meet of a subset (if exists) is the same as of its upwards-closure, so when speaking about meets of codirected subsets one may as well speak about meets of filters. Dually for joins of directed subsets / ideals (cofilters).
Bottom line - for a poset, "filtered limit" is the same as "limit of a filter".
