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nLab > Latest Changes: locally connected topos

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeFeb 10th 2010
    • (edited Nov 23rd 2012)
    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeFeb 11th 2010
    • (edited Feb 11th 2010)

    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.

    • CommentRowNumber3.
    • CommentAuthorzskoda
    • CommentTimeFeb 11th 2010

    I like it. Igor keeps mentioning some old references like Barr, Diaconescu etc. in this connection...

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeFeb 11th 2010

    Maybe you want something like C3.3.6 in the Elephant?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeFeb 12th 2010

    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.

    • CommentRowNumber6.
    • CommentAuthorzskoda
    • CommentTimeFeb 12th 2010

    why do you look at google books if I have sent you the file years ago

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeFeb 12th 2010

    Er, did you? I thought you just sent me the previous book.

    But thanks, I have it now, thanks!!

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeFeb 12th 2010

    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?

    • CommentRowNumber9.
    • CommentAuthorDavidRoberts
    • CommentTimeFeb 12th 2010
    I think so. For the case of a sheaf topos Sh(X) this is induced by the canonical map X -> pt.
    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeFeb 12th 2010
    • (edited Feb 12th 2010)

    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:

    locally n-connecteed (n,1)-topos.

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeFeb 12th 2010

    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.

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeFeb 12th 2010

    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.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeFeb 12th 2010

    Okay, thanks Todd.

    • CommentRowNumber14.
    • CommentAuthorUrs
    • CommentTimeMay 12th 2010

    as recently discussed with Mike elsewhere, I split off again

    from

    • CommentRowNumber15.
    • CommentAuthorMike Shulman
    • CommentTimeMay 12th 2010

    Thanks, looks good.

    • CommentRowNumber16.
    • CommentAuthorUrs
    • CommentTimeMay 31st 2010

    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”.

    • CommentRowNumber17.
    • CommentAuthorUrs
    • CommentTimeNov 23rd 2012

    added to the list of References at locally connected topos a pointer to the new article just out today:

    • Olivia Caramello, Site characterizations for geometric invariants of toposes, Theory and Applications of Categories, Vol. 26, 2012, No. 25, pp 710-728. (TAC)

    The same pointer could usefully be added to several other $n$Lab entries. But I am out of time now...

    • CommentRowNumber18.
    • CommentAuthorspitters
    • CommentTimeApr 30th 2018

    Every presheaf topos is locally connected.

    diff, v19, current

    • CommentRowNumber19.
    • CommentAuthorGuest
    • CommentTimeMay 11th 2022
    Hello,

    In the proof of Proposition 2.3 it is claimed that if LConst : Set --> E has a left adjoint Pi_0 : E --> Set, then an object A of E is connected in the sense of Definition 2.1 (i.e. the representable functor E(A, -) : E --> Set preserves finite coproducts) iff Pi_0(A) is a singleton set. However, in my understanding the proof of this claim that is given only shows the forward implication; it seems to me that if we know that Pi_0(A) is a singleton, then all that we can deduce is that there is a unique morphism from A to any object in the image of LConst, and it is not at all obvious (if even possible) how to deduce from this that A is connected. I believe I have shown that A will be connected if Pi_0(A) is a singleton *and* any object X of E such that Pi_0(X) is empty must be initial, but I do not believe that this latter condition is satisfied by every Grothendieck topos. So is this a mistake in the proof of Prop. 2.3, or is there some reason for why Pi_0(A) being a singleton should entail that A is connected (without any additional assumptions)?

    Thanks!
    Jason Parker
    • CommentRowNumber20.
    • CommentAuthorUrs
    • CommentTimeMay 11th 2022

    Am in a meeting with limited time, and haven’t looked at Prop. 2.3 again, but a comment on your extra condition:

    If Π 0(X)≃∅\Pi_0(X) \simeq \varnothing then the Π 0\Pi_0-unit morphism is of the form X→Π 0(X)≃∅X \xrightarrow{\;} \Pi_0(X) \simeq \varnothing and it follows that X≃∅X \simeq \varnothing since in any topos the initial object is a strict initial object.

    • CommentRowNumber21.
    • CommentAuthorUrs
    • CommentTimeMay 11th 2022
    • (edited May 11th 2022)

    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.)

    diff, v22, current

    • CommentRowNumber22.
    • CommentAuthorGuest
    • CommentTimeMay 11th 2022
    Perfect, thanks! Your edits have resolved all of my questions.

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