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92.7 Split categories fibred in groupoids

Let $S$ be a scheme contained in $\mathit{Sch}_{fppf}$. Recall that given a “presheaf of groupoids”

\[ F : (\mathit{Sch}/S)_{fppf}^{opp} \longrightarrow \textit{Groupoids} \]

we get a category fibred in groupoids $\mathcal{S}_ F$ over $(\mathit{Sch}/S)_{fppf}$, see Categories, Example 4.37.1. Any category fibred in groupoids isomorphic (!) to one of these is called a split category fibred in groupoids. Any category fibred in groupoids is equivalent to a split one.

If $F$ is a presheaf of sets then $\mathcal{S}_ F$ is fibred in sets, see Categories, Definition 4.38.2, and Categories, Example 4.38.5. The rule $F \mapsto \mathcal{S}_ F$ is in some sense fully faithful on presheaves, see Categories, Lemma 4.38.6. If $F, G$ are presheaves, then

\[ \mathcal{S}_{F \times G} = \mathcal{S}_ F \times _{(\mathit{Sch}/S)_{fppf}} \mathcal{S}_ G \]

and if $F \to H$ and $G \to H$ are maps of presheaves of sets, then

\[ \mathcal{S}_{F \times _ H G} = \mathcal{S}_ F \times _{\mathcal{S}_ H} \mathcal{S}_ G \]

where the right hand sides are $2$-fibre products. This is immediate from the definitions as the fibre categories of $\mathcal{S}_ F, \mathcal{S}_ G, \mathcal{S}_ H$ have only identity morphisms.

An even more special case is where $F = h_ X$ is a representable presheaf. In this case we have $\mathcal{S}_{h_ X} = (\mathit{Sch}/X)_{fppf}$, see Categories, Example 4.38.7.

We will use the notation $\mathcal{S}_ F$ without further mention in the following.


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