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This may seem like a stupid question, but these two concepts seem to be identified so often that it's just a detail I've overlooked. Apparently a filtered category is a generalisation of a directed set. In what way is this true?

By filtered category I mean the following: A category $\mathcal{C}$ such

1) $\mathcal{C}$ is non-empty.

2) For every pair of objects $i$ and $j$, there is an object $k$ along with arrow $\alpha: i \rightarrow k$ and $\beta : j \rightarrow k$

3) For every pair of parallel arrows $a, b: i \rightarrow j$, there is an arrow $c: j \rightarrow k$ so that $c \circ a = c \circ b$.

Suppose $\mathcal{C}$ is a small filtered category. Then surely the set $Ob(\mathcal{C})$ is a directed set where we define the order $a \leq b$ precisely when there is an arrow $a \rightarrow b$. This is clearly a preorder. Then the "directed" follows from condition (2) above. But then condition (3) seems to be superfluous and makes filtered category stronger than directed set. This goes against my understanding that it is a generalisation.

Forgive me if this is a stupid question, but is someone able to clear up the relationship between these two concepts? My experience with this is taking colimits over directed sets, such as localising a module, or find the stalk of a sheaf.

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    $\begingroup$ A cleaner definition of a directed set is that it is a p(r)oset such that any finite subset has an upper bound. The analogous cleaner definition of a filtered category, essentially a categorification, is that every finite diagram has a cocone. When we decategorify this latter definition, we recover the former as all the commutativity conditions become trivial. All diagrams commute in a p(r)oset. $\endgroup$ – Derek Elkins Oct 3 '18 at 8:08
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A directed set is a kind of poset, while a filtered category is a kind of category. Thus the key difference is that there may be unequal parallel arrows in a filtered category. The canonical example of a directed set is the set of finite, finitely generated, compact, etc. subobjects of an object. The canonical example of a filtered category is a category of finite, finitely generated, compact, etc objects.

But a directed (po)set is, when viewed as a category, certainly filtered. As a partial converse, every filtered category admits a final functor from a directed set. The surprisingly complex proof of this fact can be found in the first chapter of Adamek and Rosicky’s book on locally presentable and accessible categories.

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