The Stacks project

\begin{equation*} \DeclareMathOperator\Coim{Coim} \DeclareMathOperator\Coker{Coker} \DeclareMathOperator\Ext{Ext} \DeclareMathOperator\Hom{Hom} \DeclareMathOperator\Im{Im} \DeclareMathOperator\Ker{Ker} \DeclareMathOperator\Mor{Mor} \DeclareMathOperator\Ob{Ob} \DeclareMathOperator\Sh{Sh} \DeclareMathOperator\SheafExt{\mathcal{E}\mathit{xt}} \DeclareMathOperator\SheafHom{\mathcal{H}\mathit{om}} \DeclareMathOperator\Spec{Spec} \newcommand\colim{\mathop{\mathrm{colim}}\nolimits} \newcommand\lim{\mathop{\mathrm{lim}}\nolimits} \newcommand\Qcoh{\mathit{Qcoh}} \newcommand\Sch{\mathit{Sch}} \newcommand\QCohstack{\mathcal{QC}\!\mathit{oh}} \newcommand\Cohstack{\mathcal{C}\!\mathit{oh}} \newcommand\Spacesstack{\mathcal{S}\!\mathit{paces}} \newcommand\Quotfunctor{\mathrm{Quot}} \newcommand\Hilbfunctor{\mathrm{Hilb}} \newcommand\Curvesstack{\mathcal{C}\!\mathit{urves}} \newcommand\Polarizedstack{\mathcal{P}\!\mathit{olarized}} \newcommand\Complexesstack{\mathcal{C}\!\mathit{omplexes}} \newcommand\Pic{\mathop{\mathrm{Pic}}\nolimits} \newcommand\Picardstack{\mathcal{P}\!\mathit{ic}} \newcommand\Picardfunctor{\mathrm{Pic}} \newcommand\Deformationcategory{\mathcal{D}\!\mathit{ef}} \end{equation*}

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).


Comments (2)

Comment #2545 by Ingo Blechschmidt on

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.

Comment #2578 by on

OK, yes, here we allow both notation. It is potentially confusing, but in reality it should not be confusing because of Lemma 4.21.5. Anyway, we can probably improve the terminology. However, this is something we need to think about carefully and then fix once and for all throughout the whole Stacks project. We've already restricted the kinds of things we are allowed to do in the section discussing (co)limits over partially ordered sets... For now let's wait and see if people get confused and then maybe it will become more clear what needs to be fixed.

There are also:

  • 3 comment(s) on Section 4.19: Filtered colimits

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