Timeline for When is a basis of a topological space a Grothendieck pretopology?
Current License: CC BY-SA 4.0
14 events
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| Feb 22, 2022 at 10:37 | comment | added | მამუკა ჯიბლაძე | @DmitriPavlov Great, it is a perfect answer now I think | |
| Feb 22, 2022 at 8:20 | vote | accept | saolof | ||
| Feb 21, 2022 at 22:07 | comment | added | Dmitri Pavlov | @მამუკაჯიბლაძე: Yes, so there are at least three different ways to produce a site from a base for a topological space. I adjusted my answer to treat all three variants. | |
| Feb 21, 2022 at 22:06 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
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| Feb 21, 2022 at 22:01 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
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| Feb 21, 2022 at 21:39 | comment | added | მამუკა ჯიბლაძე | @saolof Wait sorry but I did not notice that there is a further problem with PT1! To satisfy it, it seems that one needs to define covering families in a different way: families of the form $(B_i\cap U\subseteq U)_{i\in I}$ such that each $B_i$ is basic and $U\subseteq\bigcup_iB_i$. Which is somehow less satisfying I must admit... | |
| Feb 21, 2022 at 21:02 | comment | added | Dmitri Pavlov | @saolof: The nLab claim is a mistake. There is no reason why the category of morphisms that belong to some covering family should have pullbacks, which is what this article appears to be asserting. | |
| Feb 21, 2022 at 20:58 | comment | added | saolof | OK, that sounds reasonable. I was being slightly confused by this ncatlab article which suggests that bases don't give a pretopology on the category of all open sets in X: ncatlab.org/nlab/show/Grothendieck%20pretopology#examples Is this wrong, or is there some other subtlety or choice of convention that needs to be considered? | |
| Feb 21, 2022 at 20:48 | comment | added | Dmitri Pavlov | @saolof: If you work with the ambient category of opens, then any base is a pretopology. If you work with the base itself as a category (without other opens), then the base must be closed under intersections to form a pretopology. (But in any case, the categories of sheaves of sets on both sites are equivalent, regardless of whether these sites are pretopologies.) | |
| Feb 21, 2022 at 20:47 | comment | added | Dmitri Pavlov | @მამუკაჯიბლაძე: I added the other interpretation, just in case. | |
| Feb 21, 2022 at 20:46 | history | edited | Dmitri Pavlov | CC BY-SA 4.0 |
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| Feb 21, 2022 at 19:33 | comment | added | მამუკა ჯიბლაძე | If I understand the question correctly, it asks about pretopology on the category of all open sets rather than on its subcategory formed by basic opens. In that case, if one defines covers of $U$ as families $(B_i)_{i\in I}$ of basic opens $B_i\subseteq U$ whose union is $U$, I believe PT0 is also trivially satisfied. | |
| Feb 21, 2022 at 19:27 | comment | added | saolof | So would it also be correct to say that a base forms a pretopology iff it consists of the finite intersections of a subbase? | |
| Feb 21, 2022 at 19:05 | history | answered | Dmitri Pavlov | CC BY-SA 4.0 |