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I have noticed the following error in the categorification of a local homeomorphism of spaces into a concept of a local homeomorphism of -topoi. I have a suspicion that it must have been noticed before (at least for -topoi), and if this is true, please view this discussion, in particular, as a reference request.
Background
For a topological space if is a sheaf, it has an \‘etal\‘e space which is a local homeomorphism over and sections of it are exactly the sheaf The \‘etal\‘e space construction yields an equivalence of categories
between the category of sheaves on and the category of local homeomorphisms over Moreover, assuming is sober, since sober topological spaces embed fully faithfully into topoi, for a sheaf corresponds to a geometric morphism and there turns out to be an equivalence of topoi under which this geometric morphism is equivalent to the one induced by slicing:
Using this as an example, topos theorists said that a geometric morphism of -topoi is \‘etale if it is equivalent to one of the form and this was to be a ``local homeomorphism’‘ of -topoi.
The Problem
Define an \‘etale geometric morphism of -topoi the same way, i.e.
is \‘etale if it is equivalent to one of the form
Now let be a (sober) topological space. In particular, it is a locale, hence a -topos in the sense of Lurie. Lets write this -topos by (as it is in fact the lattice of opens of ) The maps of locales that are induced by slicing are exactly those which correspond to inclusions of open sublocales. So, an \‘etale map of -topoi does not correspond to a local homeomorphism (but it is a particular kind of local homeomorphism).
You might ask: So what? Maybe this is just a defect for
No, this persists. An -topos is -localic if it is equivalent to -sheaves on an -site. There is a functor from -topoi to -topoi which is fully faithful and whose essential image is -localic -topoi, and if is an -localic, the -topos which is corresponds to is the -category of -truncated objects of
Moreover, if is an object of an then is also -localic if and only if is -truncated. So let be an object of which is -truncated but not -truncated. Then the \‘etale map of -topoi is a morphism between -localic -topoi. Let denote the -topos associated to and the one associated to Since -topoi embed fully faithfully into -topoi as -localic -topoi, this \‘etale map induced by
must correspond to a geometric morphism However, it cannot be an \‘etale map of -topoi, since the object is not -truncated. Nonetheless, such a geometric morphism is the “correct” notion of a local homeomorphism since if we let , this is a local homeomorphism of locales, in the usual sense.
Any comments? Has this been noticed for -topoi and is there a name for such geometric morphisms (and is there a characterization of them internal to -topoi?)
That’s interesting! In summary, the point is that you can’t define local homeomorphisms between n-toposes to be slice categories unless you move up to their associated $(n+1)$-toposes. This is clear when n=0, which is an essential case because it’s where the terminology ’local homeomorphism’ comes from, but it also implies that the usual definition of local homeomorphism for 1-toposes (as found, for instance, in the Elephant), is wrong. Is that an accurate summary?
I don’t think I’ve heard this mentioned before, and I’d probably remember if I had. Terminology-wise, I think I’d be in favor of using the term ’local homeomorphism’ correctly, even if it doesn’t agree with previous usage.
On the other hand, one might argue that we should define an “$n$-local-homeomorphism” to be a morphism of $\infty$-topoi that is equivalent to slicing over an $(n-1)$-truncated object. Then the existing definitions of local homeomorphism for 0-topoi and 1-topoi would both be special cases of 1-local-homeomorphisms, while the “correct” notion of local homeomorphism between $n$-topoi that you are advocating would be an $(n+1)$-local-homeomorphism. That way, the existing terminology could be maintained, according to the general principle that an unprefixed “foo” is equivalent to a “1-foo”.
As for a characterization internal to 1-topoi of the 2-local-homeomorphisms, shouldn’t they be the geometric morphisms $F\to E$ where $F$ is equivalent to the topos of internal diagrams on some internal groupoid in $E$?
As for your first comment: Yes, that’s about sums it up.
As far as your second comment, that’s exactly how I have already decided to remedy this in a paper I am writing; I define -\‘etale maps, etc. It’s nice to hear that you thought of the same solution, so I’m reassured that it was a good call.
As for your characterization, whereas I do agree that is a characterization, I was looking for one more akin to the “slice free” characterization of etale maps ( has an additional left adjoint with special properties…).
By the way, although I won’t prove it here, another way at getting at local homeomorphisms is that for a given -topos , it has a canonical topology, so we have the category of sheaves of -groupoids over (i.e. it’s associated -topos. We also have the -category There is a canonical functor given by slicing. The left Kan extension of along the yoneda embedding exists as we write it as where is the Yoneda embedding associated to any -site of definition for Local homeomorphsms are precisely those in the essential image of (This is like a topos-theoretic \‘etal\‘e space construction.)
Better yet, if you use presheaves instead of sheaves (with some size issues to deal with), you get an adjunction between presheaves and the slice category, restricting to an equivalence. (Oh yeah, is fully faithful, I should’ve said that).
I was looking for one more akin to the “slice free” characterization of etale maps
Ah. Well, I can’t think of any. (-:
BTW, you’ve probably noticed this by now, but the TeX syntax \'e
for é doesn’t work here. You can write a literal é if you have a way to input Unicode, or (if you use the “Markdown+Itex” format option, which you really should), you can write the HTML é
.
Thanks for the comment about how to typeset é. Anyhow, it should be mentioned that classically defined 'etale maps are still an important class of morphisms, because after all:
For any $n$-topos $\mathcal{E},$ $\mathcal{E} \simeq \mathfrak{Top}^{et}_n/\mathcal{E}.$
However, the theory of $n$-étale morphisms does give a nice explanation as to why $\mathfrak{Top}^{et}_n/\mathcal{E}$ is an $(n,1)$-category (since apriori one would expect it to be an $(n+1,1)$-category, as it is the slice of one). The real statement is that
$Sh_n\left(\mathcal{E}\right) \simeq \mathfrak{Top}^{n-et}_n/\mathcal{E}$where $n-et$ refers to taking only $n$-étale morphisms, and whereby $Sh_n$ is the functor $Sh_n: \mathfrak{Top}_n \hookrightarrow \mathfrak{Top}_{n+1}$ which promotes an $n$-topos to an $(n+1)$-topos. (The notation is suggestive since, at least for $n \ne \infty,$ this is equivalent to taking sheaves on $\mathcal{E}$ regarded as an $n$-site with its canonical topology. For $n=\infty,$ this is problematic however (since an $\infty$-topos is not always sheaves over itself) so by convention the functor $Sh_\infty$ is the identify, but I digress…) Now, when $n= \infty$ there is no difference between $n$-étale and étale, and $Sh_\infty=id$ so this reduces to the classical fact that etale maps over an $\infty$-topos is equivalent to the $\infty$-topos in question. When $n \ne \infty,$ then we have that $\mathcal{E} \simeq Sh_{\mathbf{n-1}}\left(\mathcal{E}\right)$ which in turn is equivalent to the full subcategory of $Sh_n\left(\mathcal{E}\right) \simeq \mathfrak{Top}^{n-et}_n/\mathcal{E}$ spanned by $(n-1)$-truncated objects, hence an $(n,1)$-category, and the $(n-1)$-truncated objects in $\mathfrak{Top}^{n-et}_n/\mathcal{E}$ are precisely the étale ones! Phew! That was a bit more complicated to spell out than I thought!
That might be the first time I’ve ever heard a theorem about $\infty$-toposes referred to as “classical”. Progress marches on, I suppose. (-;
Well, I meant more that this was a special case of $\mathcal{E} \simeq \mathfrak{Top}^{et}_n/\mathcal{E},$ (which is the “n-ification” of the classical result for ordinary topoi) when $n=\infty.$
(-:
I may be getting philosophical, but I think that maybe $\mathcal{E} \simeq \mathfrak{Top}^{et}_n/\mathcal{E}$ is the fundamental statement, and it is this equivalence which justifies the importance of étale maps. It seems that in some sense that it’s an accident that historically the fundamental statement was $Sh\left(L\right) \simeq Et/L$ for $L$ a locale, since this has a mismatch between viewing $L$ as a $1$-topos and a $0$-topos at the same time. Indeed, at $n=0$, the statement $L \simeq \mathfrak{Top}^{et}_0/L$ for a locale $L$ is an interesting and true, and the fact that this holds true for $n=1$ allows us to make the claim that $Sh(L) \simeq \mathfrak{Top}^{et}_1/Sh(L)$- and then notice that an étale morphism into a localic topos must have a localic domain and correspond to a local homeomorphisms between them. Does anyone have any thoughts?
Hi Dave,
I am certainly looking forward to seeing this stuff written up cleanly. That’ll be foundational.
@Urs: Thanks! Do you have any opinion about whether one should view étale maps being the same as the topos as foundational, or the “étalé-space construction” as foundational, as discussed in my last comment?
(Also, I am started to question as to whether or not I need to discuss all this in the paper I am writing after all, and was wondering if perhaps it should just be a separate paper entirely? Suggestions/advice? Would an OK tactic be to touch upon it, but then leave further developments for another paper?)
You could write an nLab page about it. (-;
What you say in #12 seems to me indeed to be a most useful way of thinking about the situation.
I don’t know of course exactly what paper you are about to write, but I’d think in any case in the introduction section it would be great to just sketch transparantly the situation discussed here, maybe in a kind of crisp summary as in Mike’s #2 above, followed by a paragraph with more details as in your #1.
@Urs: Thanks again, and fyi: The paper I am writing at the moment is the one establishing a general theory of higher etale stacks.
I have a hunch that the following is a characterization of local homeomorphisms (in the way I defined them) of topoi:
$f:\mathcal{E} \to \mathcal{F}$is a local homeomorphism if and only if $f$ is open and its diagonal
$\mathcal{E} \to \mathcal{E} \times_\mathcal{F} \mathcal{E}$is. Moreover, I believe that if we define a geometric morphism
$f:\mathcal{E} \to \mathcal{F}$of infinity topoi to be open if
$f:\tau_{\le -1}\mathcal{E} \to \tau_{\le -1}\mathcal{F}$is an open map of locales (this agrees with the ordinary definition of an open geometric morphism for $1$-topoi), then $f$ is 'etale if and only if $f$ and its diagonal are open. However, I haven’t thought of how to prove this yet.
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