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nLab > Latest Changes: Topos

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJul 6th 2011

    have added to Topos in the section on limits of toposes the description of the pullback of toposes by pushout of their sites of definition.

  2. I have a question about colimits in Topos\mathbf{Topos}.

    A typical topos looks like Set\mathbf{Set}, and is therefore a large category. So Topos\mathbf{Topos} is presumably a Very large category. So we might hope that Topos\mathbf{Topos} has all large colimits. But Moerdijk only states that Topos\mathbf{Topos} has all small colimits. Of course Moerdijk might be using some different set theory conventions.

    So exactly how large can colimits in Topos\mathbf{Topos} be?

  3. With logical morphisms it will definitely have large colimits, because the category is then algebraic. I am not sure about the case of geometric morphisms.
    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeOct 5th 2018

    The page Topos is about the geometric case. It’s true that ToposTopos is set-theoretically a very large 2-category, but its sub-2-category of Grothendieck toposes (which is the one that Moerdijk is talking about) is essentially only large (though not locally small), because it is equivalent to a 2-category whose objects are small sites. So it’s not reasonable to expect it to have large colimits.

    The 2-category of elementary toposes and geometric morphisms is quite ill-behaved in general; it does have some limits and colimits, but they are arguably somewhat accidental. To get good behavior you generally have to restrict to a slice category of bounded geometric morphisms over a fixed base topos, in which case things look very much again like the Grothendieck case.

  5. Thanks! I was thinking purely about the case of Grothendieck toposes and geometric morphisms, but I hadn’t realised that sites had to be small in the definition of Grothendieck topos. That make things work out much more easily.

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2025
    • (edited Jun 5th 2025)

    added (here) the statement of products of sheaf topos in ToposTopos as given by sheaves over their product sites.

    This made me realize that there must be an error in the previous discussion of pullbacks in ToposTopos (here) and I think I found and fixed it:

    Previously it said that Cat lexCat^{lex} (where the pushout of sites is to be taken for computing pullback of their sheaf toposes) is full in CatCat but, no, its morphisms are just the finitely continuous functors.

    diff, v17, current

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 5th 2025
    • (edited Jun 5th 2025)

    In https://ncatlab.org/nlab/show/Topos#AdjunctionToLocallyPresentable, shouldn’t it be ShTopos→CatShTopos \to Cat, if you want to land in locally presentable categories?

    It’s a little hard for me to tell how much of the page is really about Grothendieck toposes, or SetSet-toposes (for some fixed base topos SetSet) and bounded geometric morphisms, or what have you. (See also comment #4.)

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeJun 5th 2025

    I have adjusted the first sentence of the entry. Previously it claimed that usually the default here is “toposes” but usually its “Grothendieck toposes”

    (as in the text by Bunge & Carboni 1995 concerning that functor U:U \colon ).

    This way also the old remark (now here) makes more sense that for elementary toposes with logical functors one usually writes something else, like “LogToposLogTopos”.

    diff, v17, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeJun 6th 2025
    • (edited Jun 6th 2025)

    For illustration, I have spelled out a proof (here, so far under the assumption that the C iC_i already have finite limits) that

    PSh(C 1)× ToposPSh(C 2)≃PSh(C 1×C 2). PSh(C_1) \times_{Topos} PSh(C_2) \;\simeq\; PSh(C_1 \times C_2) \,.

    (From this the general case follows by another simple argument, but I haven’t typed that out yet.)

    diff, v19, current

1 to 9 of 9

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