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*}

Lemma 4.14.7. Suppose that $M : \mathcal{I} \to \mathcal{C}$, and $N : \mathcal{J} \to \mathcal{C}$ are diagrams whose colimits exist. Suppose $H : \mathcal{I} \to \mathcal{J}$ is a functor, and suppose $t : M \to N \circ H$ is a transformation of functors. Then there is a unique morphism

\[ \theta : \mathop{\mathrm{colim}}\nolimits _\mathcal {I} M \longrightarrow \mathop{\mathrm{colim}}\nolimits _\mathcal {J} N \]

such that all the diagrams

\[ \xymatrix{ M_ i \ar[d]_{t_ i} \ar[r] & \mathop{\mathrm{colim}}\nolimits _\mathcal {I} M \ar[d]^{\theta } \\ N_{H(i)} \ar[r] & \mathop{\mathrm{colim}}\nolimits _\mathcal {J} N } \]

commute.

Proof. Omitted. $\square$


Comments (4)

Comment #76 by Keenan Kidwell on

I think the in the diagram should be .

Comment #83 by on

Well, I am following the notation: If is a transformation of functors then is made up of maps where varies over objects of the source category of and . Since in the statement of the lemma and have source , I think the notation makes sense. I am not saying the notation is perfect though...

Comment #88 by Keenan Kidwell on

Both and have source category , right? So wouldn't writing be in accordance with the notation?

There are also:

  • 3 comment(s) on Section 4.14: Limits and colimits

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