A superextensive site is a site whose covering families can be reduced to single covers combined with stable coproducts. Many naturally occuring sites are superextensive. The prefix ‘super-’ is opposite to ‘sub-’ as in ‘subset’ and indicates that the Grothendieck topology of the site contains the extensive topology; compare subcanonical site.
Let be a finitely extensive category. A Grothendieck topology on is (finitely) superextensive if it includes the (finitely) extensive topology (generated by families of injections into finite coproducts) and every covering family is generated by a finite one.
Likewise, if is infinitary-extensive, a topology is infinitary-superextensive if it includes the infinitary-extensive topology (generated by families of injections into small coproducts) and every covering family is generated by a small one. (Of course, the last condition is vacuous if is small.)
If is a single morphism in that generates a covering family, we call it a cover. The basic property of a superextensive site is that one can replace any (finite or small, as appropriate) covering family with the single map . This is a cover, since the original covering family factors through it, and the original covering family can be written as the composite of this cover and the family of coproduct injections .
Note that covers are stable under pullback, whenever such pullbacks exist. Even without pullbacks, they always satisfy the condition that if is a cover and is any morphism, then there is a cover such that factors through . For by the pullback condition on a Grothendieck topology, there is a covering family such that each factors through , but then is the required cover. Covers are also easily seen to be stable under composition, and generate another Grothendieck topology (which is a “singleton pretopology”). If the original topology was denoted , we denote the topology generated by single covers by .
The main conclusion is:
A superextensive Grothendieck topology on an extensive category is generated by the union of and the extensive topology .
By definition, and are sub-topologies of , so it suffices to observe that any covering family in is the composite of the family of coproduct injections with the cover .
Beware, however, of the following common misconception: is not identical with . That is, although a superextensive topology is determined by its singleton covers, in the sense that it can be recovered uniquely by knowing only them, it is not generated by its singleton covers as a Grothendieck topology; we need to also include in the generating covering families.
An easy way to see that and must be different is to notice that , like any singleton pretopology, is a locally connected site, so that all constant presheaves are sheaves for . But since the initial object is covered by the empty family in , no nontrivial constant presheaf can be a sheaf for .
In general, it is not true, for two Grothendieck topologies and on the same category, that -sheafification preserves -sheaves (and hence takes them to -sheaves). For example, let be the walking arrow , let be generated by the nonidentity morphism (so that -sheaves are those for which the image of is an isomorphism), and let be generated by the empty family covering (so that -sheaves are those for which the image of is terminal).
However, if is the extensive topology and for some superextensive topology (which is then recovered as ), and coproducts preserve covers, then it is true.
Let be a finitary superextensive site, and suppose that finite coproducts of covers in are again covers. If is a presheaf on which is a -sheaf, then the -sheafification of is also a -sheaf, and hence a -sheaf.
We show that if is a -sheaf, then so is , where is the reflection into -separated presheaves. Since is -sheafification, this will prove the theorem.
Since is generated by single covers, for any we have
where the colimit is over all covers with target , and consists of those such that , where is the kernel pair of .
The extensive topology is generated by the empty covering family of the initial object, and by the injections into binary coproducts. First consider the initial object . being a sheaf for the empty family on means that is terminal?. But since is extensive, its initial object is strict, so any morphism is an isomorphism. Therefore, .
Now consider a binary coproduct . being a sheaf for this means that is an isomorphism (since coproducts are disjoint and is terminal, there is no compatibility condition to worry about). We want to show that is also an isomorphism.
First suppose given . Since covers are stable under pullback and composition, we can represent and by some for the same cover . Now since is extensive and , decomposes as , and moreover and are the pullbacks of along the two inclusions . Thus, since covers are stable under pullback, we have covers and , and so the restrictions of and to and can be represented by the restrictions of and to and respectively.
Now if and become equal in and , there must be covers for such that in for and . Now since covers are stable under coproducts, the map is a cover, and moreover and can be obtained by first restricting to and then along the two coproduct injections.
But since is a -sheaf and for , we must have that and become equal in . Now since the composite is a cover, it follows that and are equal in . Thus, is injective, i.e. is -separated.
The case of surjectivity is easier. Suppose given , represented by for some covers . Then since covers are stable under coproduct, is a cover. And since is a -sheaf, we have a which restricts to for . Hence represents some which restricts to each , as desired. Thus, is a -sheaf.
Of course, there is an analogous result for infinitary superextensive sites. With a little more work we can also prove the same thing for stacks and stackification.
This result is especially interesting because sheaves and stacks for the extensive topology are easier to come by than those for some other topologies. For instance, if is an extensive coherent category, then any internal category in represents a functor which is a stack for the extensive topology, but not usually for the coherent topology. However, the coherent topology is superextensive and its covers are precisely the regular epimorphisms, i.e. is the regular topology. Thus, the stackification of an internal category in relative to the regular topology is still a stack for the extensive topology, and hence also for the coherent topology.
The following discussion (originally at extensive category) resulted in the terminology “superextensive site.”
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 (). 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.