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92.4 Representable categories fibred in groupoids

Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. The basic object of study in this chapter will be a category fibred in groupoids $p : \mathcal{X} \to (\mathit{Sch}/S)_{fppf}$, see Categories, Definition 4.35.1. We will often simply say “let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$” to indicate this situation. A $1$-morphism $\mathcal{X} \to \mathcal{Y}$ of categories in groupoids over $(\mathit{Sch}/S)_{fppf}$ will be a $1$-morphism in the $2$-category of categories fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$, see Categories, Definition 4.35.6. It is simply a functor $\mathcal{X} \to \mathcal{Y}$ over $(\mathit{Sch}/S)_{fppf}$. We recall this is really a $(2, 1)$-category and that all $2$-fibre products exist.

Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. Recall that $\mathcal{X}$ is said to be representable if there exists a scheme $U \in \mathop{\mathrm{Ob}}\nolimits ((\mathit{Sch}/S)_{fppf})$ and an equivalence

\[ j : \mathcal{X} \longrightarrow (\mathit{Sch}/U)_{fppf} \]

of categories over $(\mathit{Sch}/S)_{fppf}$, see Categories, Definition 4.40.1. We will sometimes say that $\mathcal{X}$ is representable by a scheme to distinguish from the case where $\mathcal{X}$ is representable by an algebraic space (see below).

If $\mathcal{X}, \mathcal{Y}$ are fibred in groupoids and representable by $U, V$, then we have

92.4.0.1
\begin{equation} \label{algebraic-equation-morphisms-schemes} \mathop{Mor}\nolimits _{\textit{Cat}/(\mathit{Sch}/S)_{fppf}}(\mathcal{X}, \mathcal{Y}) \Big/ 2\text{-isomorphism} = \mathop{Mor}\nolimits _{\mathit{Sch}/S}(U, V) \end{equation}

see Categories, Lemma 4.40.3. More precisely, any $1$-morphism $\mathcal{X} \to \mathcal{Y}$ gives rise to a morphism $U \to V$. Conversely, given a morphism of schemes $U \to V$ over $S$ there exists a $1$-morphism $\phi : \mathcal{X} \to \mathcal{Y}$ which gives rise to $U \to V$ and which is unique up to unique $2$-isomorphism.


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