# The Stacks Project

## Tag 002V

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

1. the category $\mathcal{I}$ has at least one object,
2. for every pair of objects $x, y$ of $\mathcal{I}$ there exists an object $z$ and morphisms $x \to z$, $y \to z$, and
3. 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).

The code snippet corresponding to this tag is a part of the file categories.tex and is located in lines 1971–1988 (see updates for more information).

\begin{definition}
\label{definition-directed}
We say that a diagram $M : \mathcal{I} \to \mathcal{C}$ is {\it directed},
or {\it filtered} if the following conditions hold:
\begin{enumerate}
\item the category $\mathcal{I}$ has at least one object,
\item for every pair of objects $x, y$ of $\mathcal{I}$
there exists an object $z$ and morphisms $x \to z$,
$y \to z$, and
\item 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 $\mathcal{C}$.
\end{enumerate}
We say that an index category $\mathcal{I}$ is {\it directed}, or
{\it filtered} if $\text{id} : \mathcal{I} \to \mathcal{I}$ is filtered
(in other words you erase the $M$ in part (3) above).
\end{definition}

Comment #2545 by Ingo Blechschmidt on May 16, 2017 a 8:36 am UTC

Is there a clash of notation here? The chapter on limits of schemes speaks of "directed limits", by which limits over directed sets are meant (and not more general limits over filtered categories). Here, however, the terms "directed" and "filtered" are used interchangeably.

There are also 3 comments on Section 4.19: Categories.

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