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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJun 26th 2011

    at subobject classifier I have cleaned up the statement of the definition and then indicated the proof that in locally small categories subobject classifiers precisely represent the subobject-presheaf.

    • CommentRowNumber2.
    • CommentAuthorRodMcGuire
    • CommentTimeOct 6th 2016

    I added the following section because I had to search around to find some hints about when subobject classifiers don’t exist. What I added is derived from mathoverflow: Is there a finitely complete category with terminal object but NO subobject classifier?, though I didn’t add the other examples which seem mostly to do with factoring. Would it be worthwhile for someone more competent to distill that info and add it?


    Categories without subobject classifiers

    Having a subobject classifier is a vary strong property of a category and “most” categories don’t have one.

    For example, catgories with a terminal object can’t have one if there are no morphism out of the terminal object.

    • Any top bounded partial order.

    • In RingRing, the category of rings, there are no morphisms out of the terminal object the zero ring.


    (also I’m surprised the is no nLab page on the category of rings. I find it mentioned in passing at ring.

    • CommentRowNumber3.
    • CommentAuthorRodMcGuire
    • CommentTimeOct 6th 2016

    I added the following section because I had to search around to find some hints about when subobject classifiers don’t exist. What I added is derived from mathoverflow: Is there a finitely complete category with terminal object but NO subobject classifier?, though I didn’t add the other examples which seem mostly to do with factoring. Would it be worthwhile for someone more competent to distill that info and add it?


    Categories without subobject classifiers

    Having a subobject classifier is a vary strong property of a category and “most” categories don’t have one.

    For example, catgories with a terminal object can’t have one if there are no morphism out of the terminal object.

    • Any top bounded partial order.

    • In RingRing, the category of rings, there are no morphisms out of the terminal object the zero ring.


    (also I’m surprised the is no nLab page on the category of rings. I find it mentioned in passing at ring.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeOct 6th 2016

    Thanks. I edited it to say no nonidentity morphisms out of the terminal object.

    The category of rings page is called Ring, like most pages on particular categories. I added a redirect.

    • CommentRowNumber5.
    • CommentAuthorjesse
    • CommentTimeOct 6th 2016
    • (edited Oct 6th 2016)

    I added the example of abelian categories to the list. (I vaguely recall there being a slicker proof than showing that all small products of Ω\Omega with itself embed into Ω\Omega and invoking “size issues”, but I can’t recall it right now.)

    • CommentRowNumber6.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 6th 2016

    @Jesse perhaps via the notion of AT category?

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeOct 6th 2016

    Are there any interesting categories that aren’t toposes but have a subobject classifier? I suppose one could take any full subcategory of a topos closed under finite limits and containing the subobject classifier, but not closed under exponentials, such as the category of all finite or countable sets. But are there any examples that don’t occur this way?

    • CommentRowNumber8.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 6th 2016

    Mike, I think pointed sets have a subobject classifier (the obvious one with two elements).

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 6th 2016

    There are some properties of categories with a subobject classifier which rule out lots of categories; for example:

    • Such categories are balanced (mono + epi = iso). This is another way of ruling out RingRing or any poset, and also examples like PosPos, CatCat, TopTop, CMonCMon, and many others.

    • All monos are “kernels”, i.e., all are equalizers of pairs f,g:XΩf, g: X \rightrightarrows \Omega where f=(X1tΩ)f = (X \to 1 \stackrel{t}{\to} \Omega). This rules out GrpGrp without bringing in size considerations.

    • Every morphism has a unique epi-mono factorization.

    • Since Ω\Omega is internally a cartesian closed poset, all subobject orders are cartesian closed posets, and so in particular are distributive lattices if joins exist. This rules out AbAb for instance.

    Then there’s this curious little chestnut:

    • The subobject classifier has a Hopfian property, that every mono ΩΩ\Omega \to \Omega is an iso and in fact an involution (famous exercise from Johnstone’s Topos Theory).
    • CommentRowNumber10.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 6th 2016
    • (edited Oct 6th 2016)

    Returning to the issue of abelian categories, I think we can exploit distributivity of subobject lattices to generally rule these out.

    Take any object AA, so that we have three subobjects i 1:AAAi_1: A \to A \oplus A, i 2:AAAi_2: A \to A \oplus A, and Δ:AAA\Delta: A \to A \oplus A. Then i 1i 2=i_1 \vee i_2 = \top, whereas i 1Δ==i 2Δi_1 \wedge \Delta = \bot = i_2 \wedge \Delta. Under distributivity we have

    Δ=Δ=Δ(i 1i 2)=(Δi 1)(Δi 2)==\Delta = \Delta \wedge \top = \Delta \wedge (i_1 \vee i_2) = (\Delta \wedge i_1) \vee (\Delta \wedge i_2) = \bot \vee \bot = \bot

    but Δ=\Delta = \bot forces A=0A = 0. So only the trivial abelian category can have a subobject classifier.

    This probably rules out categories with biproducts generally, although some fine detail may have escaped me.

    • CommentRowNumber11.
    • CommentAuthorDavidRoberts
    • CommentTimeOct 6th 2016
    • (edited Oct 7th 2016)

    The category of classes in ZF(C) has a subobject classifier. Presumably the same is true for other boolean models of algebraic set theory.

    More generally, the class of pretoposes described in my IHES talk have a subobject classifier.

    Needless to say, all these are not locally small. (EDIT: actually, not all of them. The example constructed in my paper in Studia Logica is locally small, but it’s a topos. There are plenty more examples of that type, too)

    • CommentRowNumber12.
    • CommentAuthorTodd_Trimble
    • CommentTimeOct 7th 2016

    I’ve now added a bunch of material to subobject classifier.

    • CommentRowNumber13.
    • CommentAuthorMike Shulman
    • CommentTimeOct 7th 2016

    Thanks. I didn’t realize that a subobject classifier was an internal cartesian closed poset even if the ambient category isn’t a Heyting category. Interesting.

  1. Added weak subobject classifiers to generalisations section

    Anonymous

    diff, v50, current

    • CommentRowNumber15.
    • CommentAuthormattecapu
    • CommentTimeMar 10th 2021

    Added definition of s.o. classifier for sheaf topoi

    Matteo Capucci

    diff, v51, current

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