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*}

4.21 Limits and colimits over preordered sets

A special case of diagrams is given by systems over preordered sets.

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.

It is customary to drop the $\leq $ from the notation when talking about preordered sets, that is, one speaks of the preordered set $I$ rather than of the preordered set $(I, \leq )$. Given a preordered set $I$ the symbol $\geq $ is defined by the rule $i \geq j \Leftrightarrow j \leq i$ for all $i, j \in I$. The phrase “partially ordered set” is sometimes abbreviated to “poset”.

Given a preordered set $I$ we can construct a category: the objects are the elements of $I$, there is exactly one morphism $i \to i'$ if $i \leq i'$, and otherwise none. Conversely, given a category $\mathcal{C}$ with at most one arrow between any two objects, the set $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ is endowed with a preorder defined by the rule $x \leq y \Leftrightarrow \mathop{Mor}\nolimits _\mathcal {C}(x, y) \not= \emptyset $.

Definition 4.21.2. Let $(I, \leq )$ be a preordered set. Let $\mathcal{C}$ be a category.

  1. A system over $I$ in $\mathcal{C}$, sometimes called a inductive system over $I$ in $\mathcal{C}$ is given by objects $M_ i$ of $\mathcal{C}$ and for every $i \leq i'$ a morphism $f_{ii'} : M_ i \to M_{i'}$ such that $f_{ii} = \text{id}$ and such that $f_{ii''} = f_{i'i''} \circ f_{i i'}$ whenever $i \leq i' \leq i''$.

  2. An inverse system over $I$ in $\mathcal{C}$, sometimes called a projective system over $I$ in $\mathcal{C}$ is given by objects $M_ i$ of $\mathcal{C}$ and for every $i' \leq i$ a morphism $f_{ii'} : M_ i \to M_{i'}$ such that $f_{ii} = \text{id}$ and such that $f_{ii''} = f_{i'i''} \circ f_{i i'}$ whenever $i'' \leq i' \leq i$. (Note reversal of inequalities.)

We will say $(M_ i, f_{ii'})$ is a (inverse) system over $I$ to denote this. The maps $f_{ii'}$ are sometimes called the transition maps.

In other words a system over $I$ is just a diagram $M : \mathcal{I} \to \mathcal{C}$ where $\mathcal{I}$ is the category we associated to $I$ above: objects are elements of $I$ and there is a unique arrow $i \to i'$ in $\mathcal{I}$ if and only $i \leq i'$. An inverse system is a diagram $M : \mathcal{I}^{opp} \to \mathcal{C}$. From this point of view we could take (co)limits of any (inverse) system over $I$. However, it is customary to take only colimits of systems over $I$ and only limits of inverse systems over $I$. More precisely: Given a system $(M_ i, f_{ii'})$ over $I$ the colimit of the system $(M_ i, f_{ii'})$ is defined as

\[ \mathop{\mathrm{colim}}\nolimits _{i \in I} M_ i = \mathop{\mathrm{colim}}\nolimits _\mathcal {I} M, \]

i.e., as the colimit of the corresponding diagram. Given a inverse system $(M_ i, f_{ii'})$ over $I$ the limit of the inverse system $(M_ i, f_{ii'})$ is defined as

\[ \mathop{\mathrm{lim}}\nolimits _{i \in I} M_ i = \mathop{\mathrm{lim}}\nolimits _{\mathcal{I}^{opp}} M, \]

i.e., as the limit of the corresponding diagram.

Remark 4.21.3. Let $I$ be a preordered set. From $I$ we can construct a canonical partially ordered set $\overline{I}$ and an order preserving map $\pi : I \to \overline{I}$. Namely, we can define an equivalence relation $\sim $ on $I$ by the rule

\[ i \sim j \Leftrightarrow (i \leq j\text{ and }j \leq i). \]

We set $\overline{I} = I/\sim $ and we let $\pi : I \to \overline{I}$ be the quotient map. Finally, $\overline{I}$ comes with a unique partial ordering such that $\pi (i) \leq \pi (j) \Leftrightarrow i \leq j$. Observe that if $I$ is a directed set, then $\overline{I}$ is a directed partially ordered set. Given an (inverse) system $N$ over $\overline{I}$ we obtain an (inverse) system $M$ over $I$ by setting $M_ i = N_{\pi (i)}$. This construction defines a functor between the category of inverse systems over $I$ and $\overline{I}$. In fact, this is an equivalence. The reason is that if $i \sim j$, then for any system $M$ over $I$ the maps $M_ i \to M_ j$ and $M_ j \to M_ i$ are mutually inverse isomorphisms. More precisely, choosing a section $s : \overline{I} \to I$ of $\pi $ a quasi-inverse of the functor above sends $M$ to $N$ with $N_{\overline{i}} = M_{s(\overline{i})}$. Finally, this correspondence is compatible with colimits of systems: if $M$ and $N$ are related as above and if either $\mathop{\mathrm{colim}}\nolimits _{\overline{I}} N$ or $\mathop{\mathrm{colim}}\nolimits _ I M$ exists then so does the other and $\mathop{\mathrm{colim}}\nolimits _{\overline{I}} N = \mathop{\mathrm{colim}}\nolimits _ I M$. Similar results hold for inverse systems and limits of inverse systems.

The upshot of Remark 4.21.3 is that while computing a colimit of a system or a limit of an inverse system, we may always assume the preorder is a partial order.

Definition 4.21.4. Let $I$ be a preordered set. We say a system (resp. inverse system) $(M_ i, f_{ii'})$ is a directed system (resp. directed inverse system) if $I$ is a directed set (Definition 4.21.1): $I$ is nonempty and for all $i_1, i_2 \in I$ there exists $i\in I$ such that $i_1 \leq i$ and $i_2 \leq i$.

In this case the colimit is sometimes (unfortunately) called the “direct limit”. We will not use this last terminology. It turns out that diagrams over a filtered category are no more general than directed systems in the following sense. Before we do so we discuss the difference between

Lemma 4.21.5. Let $\mathcal{I}$ be a filtered index category. There exists a directed set $I$ and a system $(x_ i, \varphi _{ii'})$ over $I$ in $\mathcal{I}$ with the following properties:

  1. For every category $\mathcal{C}$ and every diagram $M : \mathcal{I} \to \mathcal{C}$ with values in $\mathcal{C}$, denote $(M(x_ i), M(\varphi _{ii'}))$ the corresponding system over $I$. If $\mathop{\mathrm{colim}}\nolimits _{i \in I} M(x_ i)$ exists then so does $\mathop{\mathrm{colim}}\nolimits _\mathcal {I} M$ and the transformation

    \[ \theta : \mathop{\mathrm{colim}}\nolimits _{i \in I} M(x_ i) \longrightarrow \mathop{\mathrm{colim}}\nolimits _\mathcal {I} M \]

    of Lemma 4.14.7 is an isomorphism.

  2. For every category $\mathcal{C}$ and every diagram $M : \mathcal{I}^{opp} \to \mathcal{C}$ in $\mathcal{C}$, denote $(M(x_ i), M(\varphi _{ii'}))$ the corresponding inverse system over $I$. If $\mathop{\mathrm{lim}}\nolimits _{i \in I} M(x_ i)$ exists then so does $\mathop{\mathrm{lim}}\nolimits _\mathcal {I} M$ and the transformation

    \[ \theta : \mathop{\mathrm{lim}}\nolimits _{\mathcal{I}^{opp}} M \longrightarrow \mathop{\mathrm{lim}}\nolimits _{i \in I} M(x_ i) \]

    of Lemma 4.14.8 is an isomorphism.

Proof. As explained in the text following Definition 4.21.2, we may view preordered sets as categories and systems as functors. Throughout the proof, we will freely shift between these two points of view. We prove the first statement by constructing a category $\mathcal{I}_0$, corresponding to a directed set1, and a cofinal functor $M_0 : \mathcal{I}_0 \to \mathcal{I}$. Then, by Lemma 4.17.2, the colimit of a diagram $M : \mathcal{I} \to \mathcal{C}$ coincides with the colimit of the diagram $M \circ M_0 | \mathcal{I}_0 \to \mathcal{C}$, from which the statement follows. The second statement is dual to the first and may be proved by interpreting a limit in $\mathcal{C}$ as a colimit in $\mathcal{C}^{opp}$. We omit the details.

A category $\mathcal{F}$ is called finitely generated if there exists a finite set $F$ of arrows in $\mathcal{F}$, such that each arrow in $\mathcal{F}$ may be obtained by composing arrows from $F$. In particular, this implies that $\mathcal{F}$ has finitely many objects. We start the proof by reducing to the case when $\mathcal{I}$ has the property that every finitely generated subcategory of $\mathcal{I}$ may be extended to a finitely generated subcategory with a unique final object.

Let $\omega $ denote the directed set of finite ordinals, which we view as a filtered category. It is easy to verify that the product category $\mathcal{I}\times \omega $ is also filtered, and the projection $\Pi : \mathcal{I} \times \omega \to \mathcal{I}$ is cofinal.

Now let $\mathcal{F}$ be any finitely generated subcategory of $\mathcal{I}\times \omega $. By using the axioms of a filtered category and a simple induction argument on a finite set of generators of $\mathcal{F}$, we may construct a cocone $(\{ f_ i\} , i_\infty )$ in $\mathcal{I}$ for the diagram $\mathcal{F} \to \mathcal{I}$. That is, a morphism $f_ i : i \to i_\infty $ for every object $i$ in $\mathcal{F}$ such that for each arrow $f : i \to i'$ in $\mathcal{F}$ we have $f_ i = f\circ f_{i'}$. We can also choose $i_\infty $ such that there are no arrows from $i_\infty $ to an object in $\mathcal{F}$. This is possible since we may always post-compose the arrows $f_ i$ with an arrow which is the identity on the $\mathcal{I}$-component and strictly increasing on the $\omega $-component. Now let $\mathcal{F}^+$ denote the category consisting of all objects and arrows in $\mathcal{F}$ together with the object $i_\infty $, the identity arrow $\text{id}_{i_\infty }$ and the arrows $f_ i$. Since there are no arrows from $i_\infty $ in $\mathcal{F}^+$ to any object of $\mathcal{F}$, the arrow set in $\mathcal{F}^+$ is closed under composition, so $\mathcal{F}^+$ is indeed a category. By construction, it is a finitely generated subcategory of $\mathcal{I}$ which has $i_\infty $ as unique final object. Since, by Lemma 4.17.2, the colimit of any diagram $M : \mathcal{I} \to \mathcal{C}$ coincides with the colimit of $M\circ \Pi $ , this gives the desired reduction.

The set of all finitely generated subcategories of $\mathcal{I}$ with a unique final object is naturally ordered by inclusion. We take $\mathcal{I}_0$ to be the category corresponding to this set. We also have a functor $M_0 : \mathcal{I}_0 \to \mathcal{I}$, which takes an arrow $\mathcal{F} \subset \mathcal{F'}$ in $\mathcal{I}_0$ to the unique map from the final object of $\mathcal{F}$ to the final object of $\mathcal{F}'$. Given any two finitely generated subcategories of $\mathcal{I}$, the category generated by these two categories is also finitely generated. By our assumption on $\mathcal{I}$, it is also contained in a finitely generated subcategory of $\mathcal{I}$ with a unique final object. This shows that $\mathcal{I}_0$ is directed.

Finally, we verify that $M_0$ is cofinal. Since any object of $\mathcal{I}$ is the final object in the subcategory consisting of only that object and its identity arrow, the functor $M_0$ is surjective on objects. In particular, Condition (1) of Definition 4.17.1 is satisfied. Given an object $i$ of $\mathcal{I}$, $\mathcal{F}_1, \mathcal{F}_2$ in $\mathcal{I}_0$ and maps $\varphi _1 : i \to M_0(\mathcal{F}_1)$ and $\varphi _2 : i \to M_0(\mathcal{F}_2)$ in $\mathcal{I}$, we can take $\mathcal{F}_{12}$ to be a finitely generated category with a unique final object containing $\mathcal{F}_1$, $\mathcal{F}_2$ and the morphisms $\varphi _1, \varphi _2$. The resulting diagram commutes

\[ \xymatrix{ & M_0(\mathcal{F}_{12}) & \\ M_0(\mathcal{F}_{1}) \ar[ru] & & M_0(\mathcal{F}_{2}) \ar[lu] \\ & i \ar[lu] \ar[ru] } \]

since it lives in the category $\mathcal{F}_{12}$ and $M_0(\mathcal{F}_{12})$ is final in this category. Hence also Condition (2) is satisfied, which concludes the proof. $\square$

Remark 4.21.6. Note that a finite directed set $(I, \geq )$ always has a greatest object $i_\infty $. Hence any colimit of a system $(M_ i, f_{ii'})$ over such a set is trivial in the sense that the colimit equals $M_{i_\infty }$. In contrast, a colimit indexed by a finite filtered category need not be trivial. For instance, let $\mathcal{I}$ be the category with a single object $i$ and a single non-trivial morphism $e$ satisfying $e = e \circ e$. The colimit of a diagram $M : \mathcal{I} \to Sets$ is the image of the idempotent $M(e)$. This illustrates that something like the trick of passing to $\mathcal{I}\times \omega $ in the proof of Lemma 4.21.5 is essential.

Lemma 4.21.7. If $S : \mathcal{I} \to \textit{Sets}$ is a cofiltered diagram of sets and all the $S_ i$ are finite nonempty, then $\mathop{\mathrm{lim}}\nolimits _ i S_ i$ is nonempty. In other words, the limit of a directed inverse system of finite nonempty sets is nonempty.

Proof. The two statements are equivalent by Lemma 4.21.5. Let $I$ be a directed set and let $(S_ i)_{i \in I}$ be an inverse system of finite nonempty sets over $I$. Let us say that a subsystem $T$ is a family $T = (T_ i)_{i \in I}$ of nonempty subsets $T_ i \subset S_ i$ such that $T_{i'}$ is mapped into $T_ i$ by the transition map $S_{i'} \to S_ i$ for all $i' \geq i$. Denote $\mathcal{T}$ the set of subsystems. We order $\mathcal{T}$ by inclusion. Suppose $T_\alpha $, $\alpha \in A$ is a totally ordered family of elements of $\mathcal{T}$. Say $T_\alpha = (T_{\alpha , i})_{i \in I}$. Then we can find a lower bound $T = (T_ i)_{i \in I}$ by setting $T_ i = \bigcap _{\alpha \in A} T_{\alpha , i}$ which is manifestly a finite nonempty subset of $S_ i$ as all the $T_{\alpha , i}$ are nonempty and as the $T_\alpha $ form a totally ordered family. Thus we may apply Zorn's lemma to see that $\mathcal{T}$ has minimal elements.

Let's analyze what a minimal element $T \in \mathcal{T}$ looks like. First observe that the maps $T_{i'} \to T_ i$ are all surjective. Namely, as $I$ is a directed set and $T_ i$ is finite, the intersection $T'_ i = \bigcap _{i' \geq i} \mathop{\mathrm{Im}}(T_{i'} \to T_ i)$ is nonempty. Thus $T' = (T'_ i)$ is a subsystem contained in $T$ and by minimality $T' = T$. Finally, we claim that $T_ i$ is a singleton for each $i$. Namely, if $x \in T_ i$, then we can define $T'_{i'} = (T_{i'} \to T_ i)^{-1}(\{ x\} )$ for $i' \geq i$ and $T'_ j = T_ j$ if $j \not\geq i$. This is another subsystem as we've seen above that the transition maps of the subsystem $T$ are surjective. By minimality we see that $T = T'$ which indeed implies that $T_ i$ is a singleton. This holds for every $i \in I$, hence we see that $T_ i = \{ x_ i\} $ for some $x_ i \in S_ i$ with $x_{i'} \mapsto x_ i$ under the map $S_{i'} \to S_ i$ for every $i' \geq i$. In other words, $(x_ i) \in \mathop{\mathrm{lim}}\nolimits S_ i$ and the lemma is proved. $\square$

[1] In fact, our construction will produce a directed partially ordered set.

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