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 4.21.1. Let $I$ be a set and let $\leq $ be a binary relation on $I$.

  1. We say $\leq $ is a preorder if it is transitive (if $i \leq j$ and $j \leq k$ then $i \leq k$) and reflexive ($i \leq i$ for all $i \in I$).

  2. A preordered set is a set endowed with a preorder.

  3. A directed set is a preordered set $(I, \leq )$ such that $I$ is not empty and such that $\forall i, j \in I$, there exists $k \in I$ with $i \leq k, j \leq k$.

  4. We say $\leq $ is a partial order if it is a preorder which is antisymmetric (if $i \leq j$ and $j \leq i$, then $i = j$).

  5. A partially ordered set is a set endowed with a partial order.

  6. A directed partially ordered set is a directed set whose ordering is a partial order.


Comments (4)

Comment #2102 by Keenan Kidwell on

I just noticed that in the definition of a partially ordered set, the condition of antisymmetry is omitted. Isn't it more standard to call a transitive, reflexive relation a preorder? I'm guessing the antisymmetry condition doesn't play a role in considerations involving e.g. colimits over a directed set, so this is technically moot, but I'm curious if the decision to use a (as far as I can tell) not-quite-standard definition was intentional.

Comment #2129 by on

OK, somebody else has mentioned this previously... and at the time there was a proof somewhere using the weaker notion... but I cannot find it now. The motivation for the weaker notion is that it is exactly enough conditions to turn into a category, so you can define (co)limits. I think that for almost all statements in the Stacks project it does not make a difference which definition you use. For example the proof of Lemma 4.21.5 seems to produce a partially ordered set in the stronger sense.

OK, so maybe we should change this (if you are reading this and agree please leave a comment). Then a directed set will not be a partially ordered set in general. So lot's of lemmas should get multiple statements some with directed partially ordered sets and some with just directed sets... This is lot's of work, so I will only do this if more people chime in.

Comment #2138 by Keenan Kidwell on

I was actually very happy when I saw this definition as it made me realize that the formalism of colimits and limits over directed sets worked just fine without the antisymmetry condition. I guess what it really showed me (which is something actually more standard) is that "directed sets" need not be "partially ordered" in the stronger sense. Since you don't actually seem to need the stronger sense, and it would require a non-trivial amount of work if you were to switch, I think you should leave it as it is.

Comment #2146 by on

OK, I decided to revert to the standard definition of partially ordered sets and now a directed set is a preordered set with upper bounds for finite subsets. Going through all the corrections I found that it absolutely does not matter at all! It very slightly shortens the Stacks project text because it allows us to say "directed set" instead of "directed partially ordered set" in many lemmas, propositions, etc. But I am not sure if the use of the terminology "directed set" is completely standard. I found places where people suggest one should say "directed proset" where "proset" is an abbreviation for "preordered set" but that is just plain ugly. Oh well.

Big set of changes can be found here.


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