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Some while ago we discussed using the English terms “little topos” and “big topos” when writing in English instead of the French “petit” and “gros”. I don’t recall any objections being raised, so I took the initiative to move petit topos to big and little topos as I did some expansion of it. (The floor is still open for objections, of course.)
I think it’s a good move. Along with big and little site, it makes things clear for non-French speakers (but I still think people should know a little French) and no hypothetical responses like, “the gross topos??? What on earth is that?”
I made the singular “big and little topos” a redirect. (Now your link above works! :-)
I like the entry. But maybe there is one point that needs more attention:
currently there are really two notions of big and little there, that go by the same name but are not exactly identical.
There is
topos as a space / topos as a category of spaces
topos as a space, but once over a big, once over a little site.
So the latter are really “little big toposes”. (As there are little big planets). Not sure if that’s a good thing. I believe actually that more than once in nCafe discussion some contributors did not see that we were talking about the other notion of big.
Yes, that’s right. I assume you are referring to toposes of the form E/M, where E is a big topos and M is an object of E? One could perhaps debate whether they are “little big toposes” or “big little toposes,” but they do have an odd mix of structure. They’re “little” in the sense that they “represent” a single space in some sense, but they’re “big” in the sense that their objects aren’t really locally modeled on that space; rather they’re spaces locally modeled on some fat point that just happen to be equipped with a map to the space in question. How did “gros topos” come to be used in those two different senses?
@Mike: They’re most certainly “big” toposes. The trick is that, at least in the original material, a topos is geometrically the “space” of its terminal object (this is the viewpoint emphasized in SGA4 as well as in other works of the Grothendieck school (Illusie, Giraud, and Hakim)). When we slice over a sheaf in a sheaf category, we can still see this as taking the canonical site on the category of elements and taking its category of sheaves. (This is why, for instance, we can recover the theory of schemes by looking at sheaves on Aff).
The trick is that, at least in the original material, a topos is geometrically the “space” of its terminal object (this is the viewpoint emphasized
This is the view we are talking about, not just in the original material: the topos-incarnation of any object $X$ in some topos $H$ is the over-topos $H/X$, as we just said.. So in particular $H$ itself is the topos-incarnation of its terminal object, since $H \simeq H/*$.
Yup, then it’s not at all a little topos! A little topos is the category of sheaves on the sub-site defined on “subobjects” of the terminal object (where I use this word in a totally non-strict sense). That is, take all of the covering families of the terminal object (these are just families of objects), take the full subcategory spanned by these objects, take the subsite generated by this subcategory under pullbacks, and take sheaves.
How did “gros topos” come to be used in those two different senses?
Not sure. Might be the divide between the algebraic geometry and pure topos theory community.
I am much against this nonconventional change in standard terminology. I will always write gros topos.
@Zoran: This is not our innovation. It was Johnstone’s idea.
Zoran, I don’t regard it as a change in terminology, but just as translating the original French terminology correctly into English. Like saying “sheaf” instead of “fasceaux” or “children’s drawings” instead of “dessins d’enfants.” Of course it does require the use of different words, but it has the advantage that they are actually English words, and thus much more likely to be comprehensible to an English speaker.
(I’m all in favor of people knowing French – I wish I knew more French. But I don’t think one should be obliged to know any French in order to read a paper written in English.)
Harry, the point Urs was making is that there are two different uses of big/little. What you describe in #7 is one of them; read what I wrote on the page about the other one, where “little” just means “representing a single space” instead of being a category of spaces. In that sense, E/X is little because it incarnates the space X (in a different way from Sh(X)).
I very vaguely recall that perhaps the topoi $Sh(\mathcal{O}(X))$ and $Sh(Top/X)$ are homotopic - I suppose in the sense that some ’canonical map’ induces isomorphisms on all cohomology groups/sets with local coeffs. Is this true??? It should be easy to dismiss if not.
@David:
I think what you mean to say is that the canonical map is a “homotopy equivalence”. Anyway, yes, this is stated (not sure if it’s proven) in Mac Lane - Moerdijk.
Thanks, Harry.
Thank you guys, I should look then in MacLane-Moerdijk for the statement…
I think it’s actually pretty easy to see that any local geometric morphism is a “homotopy equivalence” in whatever sense you want. In fact, any adjunction in the (∞,2)-category (∞,1)Topos should be a homotopy equivalence.
If “homotopy equivalence” means “induces an equivalence on $\Pi_\infty$,” then you just have to observe that $\Pi_\infty$ is 2-functorial, whereas it lands in the (∞,1)-category ∞Gpd where all 2-morphisms are invertible; hence adjunctions get sent to adjunctions, which are necessarily equivalences.
Since $\Pi_\infty$ represents locally constant sheaves, that means it induces an equivalence of categories of locally constant sheaves, and hence an isomorphism on cohomology with local coefficients.
On the other hand, one might want a “homotopy” of toposes to mean a geometric morphism $E\times Sh([0,1]) \to F$ like it does for classical topological spaces. I think possibly this desire is misguided, but it’s still true that any natural transformation in Topos induces a homotopy in this sense, and hence any adjunction in Topos induces a homotopy equivalence in this sense. The idea is that Topos has tensors with the interval category that are given by cartesian product with sheaves on the Sierpinski space, so a natural transformation between two geometric morphisms $E\to F$ is the same as a single geometric morphism $E\times Sh(\Sigma)\to F$. Now just pick a map $[0,1]\to \Sigma$ that doesn’t identify the endpoints, like the classifying map of $[0,\frac1 2]$.
I added a pointer to these slides by Nick Duncan to big and little toposes.
he presents a kind of definition of gros topos and discusses something related to Galois theory.
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