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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 $E$?
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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