In proposition 2.7. of the condensed notes of professors Scholze and Clausen it is said that the category of extremally disconnected sets is a site, but in the definition of a site in the Stacks Project (https://stacks.math.columbia.edu/tag/00VH) it is necessary for a site to have fibre products sometimes (Axiom (3) in the definition) and extremally disconnected sets don't have all fibre products.

The category of extremally disconnected sets is a site using the definition from the Stacks Project?

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    $\begingroup$ The Stacks Project is a work in progress and doesn't always have The Very Best things category theory has to offer (eg non-small sites that nonetheless have a good theory of sheaves!) See for now the awful temporary link: nlab-pages.s3.us-east-2.amazonaws.com/nlab/show/coverage and the definition of site there, which uses a coverage and comes from Johnstone's Sketches of an Elephant. $\endgroup$ Jan 24 at 10:00

1 Answer 1


Usage varies. Let's at least stipulate that "site" is synonymous with "category equipped with a Grothendieck topology".

Some, but not all, authors, require a site to have pullbacks, because this assumption simplifies the definition a bit. But e.g. the nlab gives the definition which doesn't assume one has pullbacks. Apparently this version of the definition goes back at least to SGA 4.

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    $\begingroup$ It should be noted that a size condition should be added in the absence of pullbacks… but the condition is automatically satisfied for essentially small sites. $\endgroup$
    – Zhen Lin
    Jan 24 at 0:15
  • $\begingroup$ @ZhenLin : although in the condensed formalism (as opposed to the pyknotic one), the site in question is not essentially small $\endgroup$ Jan 24 at 9:11
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    $\begingroup$ Indeed, that's why I mentioned it. To be explicit, the size condition I am speaking of is, I think, due to Shulman and is used to ensure that the category of small sheaves is closed under finite limits. $\endgroup$
    – Zhen Lin
    Jan 24 at 9:28

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