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nLab > Latest Changes: over-topos

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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 18th 2010
    • (edited Sep 22nd 2012)

    created over-topos

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeJan 26th 2011

    I have added to over-topos…

    • in the section etale geometric morphism statement and proof of the general case induced over any morphism in the topos;

    • in the section Slice geometric morphism statement and proof of how every geoemtric morphism lifts to one to the slice by any object of the codomain;

    • in the section Topos points statement and proof of the fact that every slice topos inherits topos points given by points of the original topos and points in the stalk of the object being sliced over.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeSep 22nd 2012
    • (edited Sep 22nd 2012)

    have added to slice topos a basic remark, right in the Definition section, about the subobject classifier in the slice.

    • CommentRowNumber4.
    • CommentAuthorDexter Chua
    • CommentTimeJul 17th 2016

    Added the construction of the power object.

    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeJul 17th 2016

    I added a remark on pullback-preserving comonads, with some references.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJul 18th 2016

    I have added to your remark a pointer to topos of coalgebras over a comonad.

    • CommentRowNumber7.
    • CommentAuthormaxsnew
    • CommentTimeJun 19th 2017
    • (edited Jun 19th 2017)

    I added a little more detail to the equivalence between a slice of a presheaf topos and the presheaf topos of the category of elements.

  8. Thanks Max! It’s good to see this spelled-out.

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2019

    We should finally give a canonical reference here for the stated fact that

    Sh(𝒞/X)≃Sh(𝒞)/y(X). Sh(\mathcal{C}/X) \simeq Sh(\mathcal{C})/y(X) \,.

    Anyone has page and verse for this at hand, either in Johnstone or in Borceux, or elsewhere?

    (Currently we keep pointing to Lurie for this statement…)

    diff, v24, current

    • CommentRowNumber10.
    • CommentAuthorAli Caglayan
    • CommentTimeMay 14th 2019

    It’s mentioned at the end of chapter VII on page 416 of Sheaves in Geometry and Logic by Mac Lane and Moerdijk. Not stated explicitly as a result though.

    • CommentRowNumber11.
    • CommentAuthorThomas Holder
    • CommentTimeMay 14th 2019

    One finds in Verdier’s exposé III.5 prop.5.4 (SGA4, p.295) the result that Sh(𝒞/X)≃Sh(𝒞)/a(X)Sh(\mathcal{C}/X)\simeq Sh(\mathcal{C})/a(X) where 𝒞\mathcal{C} is a site and aa is the associated sheaf functor. I suppose your formula is the special case that the topology that comes with 𝒞\mathcal{C} is canonical.

    • CommentRowNumber12.
    • CommentAuthorUrs
    • CommentTimeMay 14th 2019

    Thanks!!

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeMay 15th 2019

    briefly added the general statement, with the reference that Thomas gave, here.

    There is much room to expand and beautify that section. But I will leave it at that now.

    diff, v25, current

    • CommentRowNumber14.
    • CommentAuthorThomas Holder
    • CommentTimeMay 16th 2019

    Added the reference to SGA.

    diff, v26, current

    • CommentRowNumber15.
    • CommentAuthorUrs
    • CommentTimeMay 16th 2019

    Thanks!!

  16. Fixed a typo where f *f_* was used as the left adjoint to f *f^*

    Anonymous

    diff, v27, current

  17. I added a description of the category of points of a slice topos.

    Jens Hemelaer

    diff, v30, current

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeOct 8th 2021
    • (edited Oct 8th 2021)

    The beginning of the proof here was typeset in a way that made it all but unreadable. I have tried to add some more formatting. (Still room left to improve further…)

    diff, v31, current

    • CommentRowNumber19.
    • CommentAuthorUrs
    • CommentTimeOct 12th 2021
    • (edited Oct 12th 2021)

    Where the Idea-section asserted that slices of toposes being topos has “also been called” the fundamental theorem of topos theory I am making that link to to a stand-alone entry and I have added pointer to Theorem 17.4 in:

    diff, v35, current

1 to 19 of 19

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