The Stacks project

Definition 97.11.1. 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.

There are also:

  • 2 comment(s) on Section 97.11: Limit preserving

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Your email address will not be published.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 07XL. Beware of the difference between the letter 'O' and the digit '0'.