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Wrote superextensive site, with a purported proof that sheafification for the single covers does preserve extensive-sheaves in that case.
Answer to Toby's question: no, it doesn't need to be lex, I was being lazy.
Am moving the following old query-box discussion from superextensive site to here:
+–{.query}
Now you can tell me if you like my terminology. —Toby
Hmm, I’m not sure. This sort of thing does seem to crop up a lot and it might be nice to have a name for it. On the other hand, “extensive site” could potentially be misinterpreted as meaning either “any site whose underlying category is extensive,” or “an extensive category equipped with its extensive coverage,” instead of what you want, which is “an extensive category equipped with a coverage that includes its extensive coverage.” -Mike
Toby: I'm of the school that believes that when you pile on adjectives, they are allowed to interact to produce additional requirements of compatibility. (Take ’closed monoidal category’ for a pretty extreme example.) One can say ’trivial extensive site’ for an extensive category equipped with the coverage of coproduct inclusions.
It's hard to use Google to determine if ’extensive site’ already has a meaning.
Mike: I would argue that in “closed monoidal category” the adjective “closed” modifies “monoidal category” rather than simply “category.” However, I don’t dispute the general principle. I think there is more potential for confusion with “extensive site” than there is with “closed monoidal category,” however, but as I said I’m ambivalent at the moment.
Toby: And similarly, ’extensive’ modifies ’site’, not ’category’ (which doesn't actually appear, so this practice can only be more justified). (^_^)
Mike: I am now firmly convinced that we should not use “extensive site” for this, because it seems to be an otherwise universal convention that the same adjective is applied to the coverage and the site. For example, in A2.1.11(b) the Elephant uses “coherent site” for “coherent category equipped with its coherent coverage,” without even formally defining “coherent site” (only “coherent coverage”). And I believe algebraic geometers speak of “the Zariski site” or “the etale site” meaning some appropriate category equipped with the Zariski coverage or the etale coverage. So I believe “extensive site” should really mean “an extensive category equipped with its extensive coverage.”
Toby: And here I was thinking that I should complain about this very language! I don't know about coherent sites, but certainly there are plenty of extensive sites (in the general sense) that aren't extensive sites (in the restricted sense), including what is arguably the original large site ($\Top$). If you want the restricted sense, then you only have to add an adjective (’trivial’), but what do you do if you want the general sense?
But perhaps you have an answer for that?
Mike: You can certainly complain about it, but I think it one of those numerous examples of well-established terminology that is suboptimal, but not bad enough to be worth trying to change. Changing any piece of terminology is hard enough, and insisting that everyone change the words they are using for familiar concepts is part of what originally gave category theory a bad name in the U.S., and has resulted in at least one unreadable book which might otherwise have been quite good. Also, as Jaap van Oosten pointed out in the introduction to his book “Realizability,” the only thing worse than bad terminology is continually changing terminology. I’m glad that “triple” was changed to “monad,” because “triple” was truly a horrible term, but the fact that it was changed still makes it more difficult to read old papers. So I think we need to pick and choose our terminological battles very carefully.
One can, of course, always say “a site whose underlying category is extensive.” By analogy with “subcanonical coverage” (= contained in the canonical coverage) one might also say “superextensive coverage.” And despite how common such sites may be, it’s not immediately clear to me how important it is to identify them.
Toby: They're important for a paper that I'm working on with John, if only to link to previous work. But ’superextensive site’ will do.
Mike: Okay, I changed the text. If we’re agreed on that terminology, we could probably remove this discussion. Or maybe we should collect all our past terminological arguments on one page, so that people who are interested can see what we went through?
Toby: Should we create a page terminological disputes (for archiving only, with a note of the relevant original page if anyone wants to reopen discussion) or use (or ask for) a Café post. The former is more convenient, the latter arguably more proper.
=–
Somewhat silly question: what’s a good name for sites where the empty sieve covers initial objects? This is the limiting case for $\kappa$-ary superextensive sites at $\kappa = 1$. This is subcanonical if and only if the category has a strict initial object, which one might think of as being a $\kappa$-ary extensive category for $\kappa = 1$…
If you have a strict initial object, then ’nullary superextensive’ seems fine to me. If not, I’m not sure.
Continuing on the same probably misguided theme, what should the definition of superextensive topology be for non-extensive categories? In contrast to the $\kappa = 1$ case, declaring coproduct cocones to be covering doesn’t force the Yoneda embedding into sheaves to preserve them; we also need to assume that coproducts are disjoint unions. In particular, even if we assume the existence of coproducts, the canonical topology being superextensive may not imply that coproducts are disjoint unions; but if we assume that coproducts are disjoint unions, then canonical-is-superextensive implies that coproducts are preserved by pullback.
What about only taking as covering families coproduct cocones for disjoint coproducts? Proposition 2 at that page seems to imply that such families form a coverage where the squares required to exist are cartesian.
I think the right definition (assuming the existence of $\kappa$-ary coproducts, but no more) is that (1) the coprojections of these coproducts form covering families, and (2) these coproducts are “locally weakly disjoint” in the sense that the squares that are required to be pullbacks for a disjoint coproduct are instead “local prelimits” in the sense of my exact-completions paper.
That makes sense. I guess the point is to make it a theorem that the Yoneda embedding into superextensive sheaves preserves coproducts?
I suppose that’s part of the point.
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