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*}

4.17 Cofinal and initial categories

In the literature sometimes the word “final” is used instead of cofinal in the following definition.

Definition 4.17.1. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor between categories. We say $\mathcal{I}$ is cofinal in $\mathcal{J}$ or that $H$ is cofinal if

  1. for all $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{J})$ there exists a $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and a morphism $y \to H(x)$, and

  2. given $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{J})$, $x, x' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and morphisms $y \to H(x)$ and $y \to H(x')$ there exists a sequence of morphisms

    \[ x = x_0 \leftarrow x_1 \rightarrow x_2 \leftarrow x_3 \rightarrow \ldots \rightarrow x_{2n} = x' \]

    in $\mathcal{I}$ and morphisms $y \to H(x_ i)$ in $\mathcal{J}$ such that the diagrams

    \[ \xymatrix{ & y \ar[ld] \ar[d] \ar[rd] \\ H(x_{2k}) & H(x_{2k + 1}) \ar[l] \ar[r] & H(x_{2k + 2}) } \]

    commute for $k = 0, \ldots , n - 1$.

Lemma 4.17.2. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor of categories. Assume $\mathcal{I}$ is cofinal in $\mathcal{J}$. Then for every diagram $M : \mathcal{J} \to \mathcal{C}$ we have a canonical isomorphism

\[ \mathop{\mathrm{colim}}\nolimits _\mathcal {I} M \circ H = \mathop{\mathrm{colim}}\nolimits _\mathcal {J} M \]

if either side exists.

Proof. Omitted. $\square$

Definition 4.17.3. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor between categories. We say $\mathcal{I}$ is initial in $\mathcal{J}$ or that $H$ is initial if

  1. for all $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{J})$ there exists a $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and a morphism $H(x) \to y$,

  2. for any $y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{J})$, $x , x' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ and morphisms $H(x) \to y$, $H(x') \to y$ in $\mathcal{J}$ there exists a sequence of morphisms

    \[ x = x_0 \leftarrow x_1 \rightarrow x_2 \leftarrow x_3 \rightarrow \ldots \rightarrow x_{2n} = x' \]

    in $\mathcal{I}$ and morphisms $H(x_ i) \to y$ in $\mathcal{J}$ such that the diagrams

    \[ \xymatrix{ H(x_{2k}) \ar[rd] & H(x_{2k + 1}) \ar[l] \ar[r] \ar[d] & H(x_{2k + 2}) \ar[ld] \\ & y } \]

    commute for $k = 0, \ldots , n - 1$.

This is just the dual notion to “cofinal” functors.

Lemma 4.17.4. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor of categories. Assume $\mathcal{I}$ is initial in $\mathcal{J}$. Then for every diagram $M : \mathcal{J} \to \mathcal{C}$ we have a canonical isomorphism

\[ \mathop{\mathrm{lim}}\nolimits _\mathcal {I} M \circ H = \mathop{\mathrm{lim}}\nolimits _\mathcal {J} M \]

if either side exists.

Proof. Omitted. $\square$

Lemma 4.17.5. Let $F : \mathcal{I} \to \mathcal{I}'$ be a functor. Assume

  1. the fibre categories (see Definition 4.31.2) of $\mathcal{I}$ over $\mathcal{I}'$ are all connected, and

  2. for every morphism $\alpha ' : x' \to y'$ in $\mathcal{I}'$ there exist a morphism $\alpha : x \to y$ in $\mathcal{I}$ such that $F(\alpha ) = \alpha '$.

Then for every diagram $M : \mathcal{I}' \to \mathcal{C}$ the colimit $\mathop{\mathrm{colim}}\nolimits _\mathcal {I} M \circ F$ exists if and only if $\mathop{\mathrm{colim}}\nolimits _{\mathcal{I}'} M$ exists and if so these colimits agree.

Proof. One can prove this by showing that $\mathcal{I}$ is cofinal in $\mathcal{I}'$ and applying Lemma 4.17.2. But we can also prove it directly as follows. It suffices to show that for any object $T$ of $\mathcal{C}$ we have

\[ \mathop{\mathrm{lim}}\nolimits _{\mathcal{I}^{opp}} \mathop{Mor}\nolimits _\mathcal {C}(M_{F(i)}, T) = \mathop{\mathrm{lim}}\nolimits _{(\mathcal{I}')^{opp}} \mathop{Mor}\nolimits _\mathcal {C}(M_{i'}, T) \]

If $(g_{i'})_{i' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I}')}$ is an element of the right hand side, then setting $f_ i = g_{F(i)}$ we obtain an element $(f_ i)_{i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})}$ of the left hand side. Conversely, let $(f_ i)_{i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})}$ be an element of the left hand side. Note that on each (connected) fibre category $\mathcal{I}_{i'}$ the functor $M \circ F$ is constant with value $M_{i'}$. Hence the morphisms $f_ i$ for $i \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I})$ with $F(i) = i'$ are all the same and determine a well defined morphism $g_{i'} : M_{i'} \to T$. By assumption (2) the collection $(g_{i'})_{i' \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{I}')}$ defines an element of the right hand side. $\square$

Lemma 4.17.6. Let $\mathcal{I}$ and $\mathcal{J}$ be a categories and denote $p : \mathcal{I} \times \mathcal{J} \to \mathcal{J}$ the projection. If $\mathcal{I}$ is connected, then for a diagram $M : \mathcal{J} \to \mathcal{C}$ the colimit $\mathop{\mathrm{colim}}\nolimits _\mathcal {J} M$ exists if and only if $\mathop{\mathrm{colim}}\nolimits _{\mathcal{I} \times \mathcal{J}} M \circ p$ exists and if so these colimits are equal.

Proof. This is a special case of Lemma 4.17.5. $\square$


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