The Stacks project

Lemma 94.18.2. Suppose $\mathit{Sch}_{fppf}$ is contained in $\mathit{Sch}'_{fppf}$. Let $S$ be an object of $\mathit{Sch}_{fppf}$. Denote $\textit{Algebraic-Stacks}/S$ the $2$-category of algebraic stacks over $S$ defined using $\mathit{Sch}_{fppf}$. Similarly, denote $\textit{Algebraic-Stacks}'/S$ the $2$-category of algebraic stacks over $S$ defined using $\mathit{Sch}'_{fppf}$. The rule $\mathcal{X} \mapsto f^{-1}\mathcal{X}$ of Lemma 94.18.1 defines a functor of $2$-categories

\[ \textit{Algebraic-Stacks}/S \longrightarrow \textit{Algebraic-Stacks}'/S \]

which defines equivalences of morphism categories

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Algebraic-Stacks}/S}(\mathcal{X}, \mathcal{Y}) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{Algebraic-Stacks}'/S}(f^{-1}\mathcal{X}, f^{-1}\mathcal{Y}) \]

for every objects $\mathcal{X}, \mathcal{Y}$ of $\textit{Algebraic-Stacks}/S$. An object $\mathcal{X}'$ of $\textit{Algebraic-Stacks}'/S$ is equivalence to $f^{-1}\mathcal{X}$ for some $\mathcal{X}$ in $\textit{Algebraic-Stacks}/S$ if and only if it has a presentation $\mathcal{X} = [U'/R']$ with $U', R'$ isomorphic to $f^{-1}U$, $f^{-1}R$ for some $U, R \in \textit{Spaces}/S$.

Proof. The statement on morphism categories is a consequence of the more general Stacks, Lemma 8.12.12. The characterization of the “essential image” follows from the description of $f^{-1}$ in the proof of Lemma 94.18.1. $\square$

Comments (2)

Comment #5524 by Olivier de Gaay Fortman on

I think you mean by Algebraic-Stacks/S the 2-category of algebraic stacks over , defined using , and similarly for Algebraic-Stacks'/S.

Comment #5717 by on

Of course, I did not think anybody would ever look at this section. Thanks very much! The fix is here.

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View Lemma 94.18.2 as pdf