94.4 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 fibred 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
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
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.
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (1)
Comment #6529 by Hadi Hedayatzadeh on