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There seems to be a pervasive myth that a -topos is the same thing as a locale. (The latest example was added today to (0,1)-category.) This is wrong; not every -topos is a locale. Only the Grothendieck -toposes are locales. (See (0,1)-topos.)
Yes, sure. This is a problem of terminology induced from the problem that currently we say -topos throughout for sheaf -topos .
Why do we do that? Why not “Grothendieck -topos”?
Some reasons
laziness
I’d rather say “-sheaf -topos”. Terms should be descriptive.
in either case, it gets quite tiring to keep the qualifier around.
Lurie does it that way
There is currently no theory of elementary -toposes, so I’d be inclined to have the -sheaf case as default, and start saying elementary -topos when that becomes necessary.
None of these is a particularly good reason, of course. Maybe a solution would be to create a standard warning box about terminology and include it into all relevant entries. That, however, brings me back to the first point.
I should add that I did and do add the “-sheaf”-qualifier when I think of it. But sometimes I forget.
Keep in mind that the original meaning of “topos” was a sheaf topos! Only later did people come along and want to say plain “topos” for the elementary version and add the adjective “sheaf” or “Grothendieck” for the original version. I have no problem with using plain “topos” to mean either version, if the context makes it clear which is meant; and if the context doesn’t make it clear I think an adjective should be added in either case.
Moreover, since the appropriate notion of “elementary n-topos” is not entirely clear for any , I think it is more excusable to say “-topos” to mean the Grothendieck version in that case. I know we sometimes say an elementary (0,1)-topos is a Heyting algebra, but it’s not really clear to me that a Heyting algebra deserves to be called a (0,1)-topos rather than a (0,1)--pretopos, since there is no (0,1)-analogue of a subobject classifer.
Urs 4.2: as you know, Grothendieck topos is only equivalent to a sheaf topos, it does not need to be set-theoretically consisting of sheaves, so I would not quite replace the name by the sheaf topos. I think that the distinction is sometimes useful because it reminds of the fact that the Grothendieck topoi can be abstractly recognized (or defined) by Giraud’s theorem in one of the variants. If I were teaching words sheaf topos, why would I call it topos at all. It is a category of sheaves, period. But once I call it topos, I assume the setting with Giraud’s theorem and if I introduce this word I owe some explanation to the students, saying that Grothendieck introduced topoi for that and that reason, as the category has essential cohomological information on the underlying space/site bla bla and that it is further confirmed by Giraud’s theorem.
Mike 6: interesting remarks.
I personally do not think that topos without modifier is by default elementary. By default it is ambiguous without modifier – for some traditions and books the topic is the Grothendieck version, for some the elementary version. The fact that the latter subsumes the first is not an argument against the linguistic fact that a part of the community by default take Grothendieck as default and a part elementary as the default.
The term “sheaf -topos” is not ideal, either. But “Grothendieck -topos” sounds worse to me. If you name this after a person, you should name it a “Rezk -topos” or “Rezk-Lurie -topos”. But I have to agree with the foreword of Moerdijk-Reyes that in the long run it is wise not to name concepts after people. It is better to think of descriptive terms.
I have seen such long names like Grothendieck-Rezk-Lurie terms, I think at MathOverflow…Is Rezk’s version also in terms of quasicategories ? I mean the Gabriel-Vezzosi treatment of infinity-topoi is after Rezk’s and before Lurie’s treatment and is in terms of Segal categories.
Joyal says in his book -topos for quasicategory variant where is his notation, I think, for the coherent nerve of the large quasicategory of simplicially enriched groupoids (like the sheaf topos over a point).
It is surely OK to have new names for infinity notions (though, it looks to me, that in other cases it was against your principle, where you like to call them by the same name, like infinity sheaf and not infinity stack).
I have seen such long names like Grothendieck-Rezk-Lurie terms, I think at MathOverflow…
I write this on the Lab when I conciously try to avoid the discussion that we are having now. See here.
Is Rezk’s version also in terms of quasicategories ?
He did it in terms of “model toposes”, but he was the one who identified the “-Giraud axiom” characterization in that language.
In that case, Toen-Vezzosi 2002 qualify for the name as well (as far as Giraud they have an extensive treatment of Giraud’s theorem in addition to the sheaf approach). Right, I recall now of seeing Rezk’s model theoretic approach. The ideas of quasicategory version presented in Joyal attribute ideas to (the digestion) of Rezk, so it may be that Joyal was the first to look at U-topoi in that setup.
Okay, so we should speak of
Grothendieck-Toën-Vezzosi-Rezk-Joyal-Lure -toposes
;-)
I have seen that version before as well :)
I agree that descriptive names, rather than personal names, are good. I have no problem with calling something a “sheaf topos” if it is only equivalent to a category of sheaves, just as I have no problem with calling a set “the natural numbers” even if its elements are not von Neumann ordinals.
I also agree, if it was not clear from my first comment, that “topos” should be regarded as ambiguous without modifier. Moreover, the fact that every sheaf topos is an elementary topos is only true if you assume to start out with that Set is an elementary topos! If you work in predicative mathematics, where Set is only a pretopos, then a lot of sheaf-topos-theory works just as well, but the “sheaf toposes” (i.e. categories of sheaves) one gets are not, of course, elementary toposes.
I agree completely with Mike #14’s first paragraph; it’s just what I wanted to say. And I’m coming around to the idea that we should never say “topos” (in general context) without an adjective.
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