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After the third time typing "pretopology" into my nlab-goto box and ending up at pretopological space instead of where I wanted to be, namely Grothendieck pretopology, I changed the redirect. It seems likely to me that the latter notion will be of more interest to more of our clientele than the former. But if you disagree, speak up.
I also added a Wikipedia-style "see also" note to the top of Grothendieck pretopology. Should we do that sort of thing in general?
Yes, I agree with your re-redirection.
I'm not sure about the hatnote, but we can try it and see if it annoys us.
Here's a slightly more questionable re-redirection I'm tempted to make: I would like to make regular topology point to regular category rather than regular space. Thoughts?
Yeah, that is more questionable. I would rather have people write regular coverage; the generic term pretopology is one thing, but I'm inclined to deprecate those terms for the most part. (Although a pretopology and a precoverage aren't quite the same thing, any specific Grothendieck topology, such as the regular one on a regular category, may be specified by a pretopology, a precoverage, a coverage, or whatever). Whereas the (albeit probably less commonly used on the Lab) term from general topology really has no alternative.
I don't think I've ever heard anyone talk about a "regular topology" with reference to a regular topological space; I've always only heard the adjective applied to the space, not the topology. But I'm not all that familiar with point-set topology either.
Regarding "topology" versus "coverage," I know I was once very much in favor of the latter, and I still think it's better in an absolute sense, but I'm getting the feeling that there are so many algebraic geometers using "topology" that there's basically no chance of changing it in mathematics at large, and using different terminology than everyone else is only going to make it more difficult to make connections between category theorists and other mathematicians.
I don't think I've ever heard […].
I'm sure that I have, certainly for items lower on the scale like T1. When I get back home, I'll be able to cite examples. In the meantime, check out this usage (start just after the bullet points); we definitely want topologists to say that, and it's the nicest phrasing for them.
Would the algebraic geometers not know what the ‘regular coverage’ is?
Okay, fair enough; we may have to just accept the conflict. But I still think that regular Grothendieck topologies are more likely to be of interest to most nlabbers than regular topological spaces.
Would the algebraic geometers not know what the ‘regular coverage’ is?
Well, maybe, if they know what a "coverage" is, but I don't think most of them do. (They'd also of course have to know what a regular category is, but that's also true for "regular topology.") It's unclear to me how many of them have even heard of the Elephant, let alone read enough of it to know about its terminological quirks.
I have created induced topology and subspace with intention to develop them separately inspite of partial overlap. I think that the present content already shows the natural differences.
I added a note on more general kinds of induced topology, inviting us to write about them at weak topology.
Great. I also read (I think George Whitehead used it) "coherent topology".
I dislike that there are holidays these days right now when I am in a hi mood to go to work, read (and do more on nlab). But if I do not use the opportunity when others are free I will miss the family, friends, foofd, going to the nature etc. in the days to follow...
Mike made a lnik to sink at induced topology, and I filled it. There is a terminological difficulty here that people may want to look at.
Yeah, I don't know what to do about it. "Cosink" seems like the best of a bad set of options at the moment.
Another possibility is to strictly say source object for source/domain. But redirecting source itself to sink would break a lot of links at the nLab, so I'm inclined to say cosink there at least.
The page Grothendieck pretopology used to claim that if we drop the first and third conditions, we end up with a coverage. Actually what we obtain is a slightly stronger notion where the covering families are actually closed under pullbacks, which at the page coverage is called a “cartesian coverage”. I fixed the reference accordingly.
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