\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*}
89 Criteria for Representability
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Section 89.1: Introduction
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Section 89.2: Conventions
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Section 89.3: What we already know
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Section 89.4: Morphisms of stacks in groupoids
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Section 89.5: Limit preserving on objects
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Section 89.6: Formally smooth on objects
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Section 89.7: Surjective on objects
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Section 89.8: Algebraic morphisms
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Section 89.9: Spaces of sections
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Section 89.10: Relative morphisms
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Section 89.11: Restriction of scalars
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Section 89.12: Finite Hilbert stacks
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Section 89.13: The finite Hilbert stack of a point
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Section 89.14: Finite Hilbert stacks of spaces
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Section 89.15: LCI locus in the Hilbert stack
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Section 89.16: Bootstrapping algebraic stacks
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Section 89.17: Applications
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Section 89.18: When is a quotient stack algebraic?
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Section 89.19: Algebraic stacks in the étale topology