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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.
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?
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 $Ring$, 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.
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?
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 $Ring$, 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.
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.
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 $\Omega$ with itself embed into $\Omega$ and invoking “size issues”, but I can’t recall it right now.)
@Jesse perhaps via the notion of AT category?
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?
Mike, I think pointed sets have a subobject classifier (the obvious one with two elements).
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 $Ring$ or any poset, and also examples like $Pos$, $Cat$, $Top$, $CMon$, and many others.
All monos are “kernels”, i.e., all are equalizers of pairs $f, g: X \rightrightarrows \Omega$ where $f = (X \to 1 \stackrel{t}{\to} \Omega)$. This rules out $Grp$ without bringing in size considerations.
Every morphism has a unique epi-mono factorization.
Since $\Omega$ 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 $Ab$ for instance.
Then there’s this curious little chestnut:
Returning to the issue of abelian categories, I think we can exploit distributivity of subobject lattices to generally rule these out.
Take any object $A$, so that we have three subobjects $i_1: A \to A \oplus A$, $i_2: A \to A \oplus A$, and $\Delta: A \to A \oplus A$. Then $i_1 \vee i_2 = \top$, whereas $i_1 \wedge \Delta = \bot = i_2 \wedge \Delta$. Under distributivity we have
$\Delta = \Delta \wedge \top = \Delta \wedge (i_1 \vee i_2) = (\Delta \wedge i_1) \vee (\Delta \wedge i_2) = \bot \vee \bot = \bot$but $\Delta = \bot$ forces $A = 0$. 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.
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)
I’ve now added a bunch of material to subobject classifier.
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.
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