The Stacks project

93.7 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.

Comments (0)

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Your email address will not be published.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).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 04SU. Beware of the difference between the letter 'O' and the digit '0'.