## Tag `002V`

Chapter 4: Categories > Section 4.19: Filtered colimits

Definition 4.19.1. We say that a diagram $M : \mathcal{I} \to \mathcal{C}$ is

directed, orfilteredif the following conditions hold:

- the category $\mathcal{I}$ has at least one object,
- for every pair of objects $x, y$ of $\mathcal{I}$ there exists 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, orfilteredif $\text{id} : \mathcal{I} \to \mathcal{I}$ 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}
```

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