The Stacks project

Definition 98.11.1. Let $S$ be a scheme. Let $\mathcal{X}$ be a category fibred in groupoids over $(\mathit{Sch}/S)_{fppf}$. We say $\mathcal{X}$ is limit preserving if for every affine scheme $T$ over $S$ which is a limit $T = \mathop{\mathrm{lim}}\nolimits T_ i$ of a directed inverse system of affine schemes $T_ i$ over $S$, we have an equivalence

\[ \mathop{\mathrm{colim}}\nolimits \mathcal{X}_{T_ i} \longrightarrow \mathcal{X}_ T \]

of fibre categories.

Comments (2)

Comment #6246 by DatPham on

Maybe this is trivial to ask but is there a definition for the notion (i.e. a colimit of groupoids)? My understanding is that this is just a suggestive notation (and the meaning of the equivalence is explained in the paragraph followed the above definition). Is this correct or I am missing something?

Comment #6247 by on

First of all: yes, please see the explanation following the definition for what the definition means. OTOH you can define colimits of directed systems of groupoids exactly as suggested by this explanation and for the purposes of the Stacks project, this is the correct definition. You can also define if you just assume: (1) is a directed set (Definition 4.21.1), (2) for each we have a groupoid , and (3) for each we have a functor with , and (4) for every a transformation such that this data forms a pseudo functor from to the -category of groupoids, see Definition 4.29.5. The colimit of such a beast constructed more or less in the same way will have a weak universal property which I leave it up to you to clarify.

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  • 2 comment(s) on Section 98.11: Limit preserving

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