Definition 4.19.1. We say that a diagram $M : \mathcal{I} \to \mathcal{C}$ is *directed*, or *filtered* if the following conditions hold:

the category $\mathcal{I}$ has at least one object,

for every pair of objects $x, y$ of $\mathcal{I}$ there exist an object $z$ and morphisms $x \to z$, $y \to z$, and

for every pair of objects $x, y$ of $\mathcal{I}$ and every pair of morphisms $a, b : x \to y$ of $\mathcal{I}$ there exists a morphism $c : y \to z$ of $\mathcal{I}$ such that $M(c \circ a) = M(c \circ b)$ as morphisms in $\mathcal{C}$.

We say that an index category $\mathcal{I}$ is *directed*, or *filtered* if $\text{id} : \mathcal{I} \to \mathcal{I}$ is filtered (in other words you erase the $M$ in part (3) above).

## Comments (2)

Comment #2545 by Ingo Blechschmidt on

Comment #2578 by Johan on

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