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started locally connected topos
I wanted to type a proof at locally connected topos that not only does for every locally connetced topos exist a left adjoint to the constant sheaf functor, but that conversely when that constant sheaf functor has a left adjoint, the topos is locally connected.
It seemed pretty obvious, but while typing a would-be proof I realized that either either I am missing something or it is more subtle.
I like it. Igor keeps mentioning some old references like Barr, Diaconescu etc. in this connection...
Maybe you want something like C3.3.6 in the Elephant?
Yes!
Thanks, I should have looked there earlier.
So what I am after at homotopy groups in an (infinity,1)-topos, as I learn now, is the notion of essential geometric morphism (to be created in a second)!
I have to access the Elephant through Google-books at the moment. Of course, the preview breaks off right after page 651, with C.3.3.6 being on 652.
Could you just briefly say what the statement there is? That would be much appreciated.
why do you look at google books if I have sent you the file years ago
Er, did you? I thought you just sent me the previous book.
But thanks, I have it now, thanks!!
I need to learn the notion and theory of toposes "indexed over" other toposes.
I suppose for the case where I am geometrically mapping into Set I can ignore this for the moment?
Thanks, David.
Okay, I have added that theorem to locally connected topos.
This allowed me finally to say confidently what I had indended to say all along:
Who wrote that query box?
Why not just make the changes, instead of trying to make me make the changes.
Conversely, by the way, I think we should go through plenty of pages on properties of topological spaces and link them to their topos analogs.
We really need a big table of contents here eventually. I think we should start a big TOC at Elephant indexing the entire book. Eventually.
Urs, that was me who wrote the query box. I made a note of it in a separate discussion, which was my error. I'm not trying to make you make the changes, I just wanted the comment to be seen, rather than just make the change and maybe have it go unnoticed unless one checks Recent Changes. I'll make the change now. Sorry if I irritated you.
Okay, thanks Todd.
as recently discussed with Mike elsewhere, I split off again
from
Thanks, looks good.
Started a section Examples at locally connected topos, but am being interrupted.
Wanted to point out somewhere that diffeological spaces form a “locally contractible quasi-topos”.
added to the list of References at locally connected topos a pointer to the new article just out today:
The same pointer could usefully be added to several other $n$Lab entries. But I am out of time now...
Am in a meeting with limited time, and haven’t looked at Prop. 2.3 again, but a comment on your extra condition:
If $\Pi_0(X) \simeq \varnothing$ then the $\Pi_0$-unit morphism is of the form $X \xrightarrow{\;} \Pi_0(X) \simeq \varnothing$ and it follows that $X \simeq \varnothing$ since in any topos the initial object is a strict initial object.
Have now edited this into the proof (here), also tried to polish up verbiage and formatting throughout.
(I don’t claim that the entire proof couldn’t be replaced by a quicker one – that’s in part what the pointer to Johnstone is about – have just tried to polish up the proof the way it was laid out.)
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