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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 $Ring$, 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.

• 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 $Ring$, 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.

• 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 $\Omega$ with itself embed into $\Omega$ 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 7th 2016

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

• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeOct 7th 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 $Ring$ 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:

• The subobject classifier has a Hopfian property, that every mono $\Omega \to \Omega$ is an iso and in fact an involution (famous exercise from Johnstone’s Topos Theory).
• CommentRowNumber10.
• CommentAuthorTodd_Trimble
• CommentTimeOct 7th 2016
• (edited Oct 7th 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 $A$, 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.

• CommentRowNumber11.
• CommentAuthorDavidRoberts
• CommentTimeOct 7th 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

• CommentRowNumber15.
• CommentAuthormattecapu
• CommentTimeMar 10th 2021

Added definition of s.o. classifier for sheaf topoi

Matteo Capucci

• CommentRowNumber16.
• CommentAuthorThomas Holder
• CommentTimeMay 4th 2022

Added a reference to the paper

• Peter Johnstone, Collapsed Toposes and Cartesian Closed Varieties , JA 129 (1990) pp.446-480.

where he discusses subobject classifiers in varieties.

• CommentRowNumber17.
• CommentAuthorSeth Chaiken
• CommentTimeAug 7th 2022

In the first definition,

there is a unique morphism χ U:X→Ω such that there is a pullback diagram of the following form:

I think it’s more clear to say: there is a unique morphism χ U:X→Ω such that the following is a pullback diagram:

(Also the period after the diagram should be removed, since the sentence was ended by the colon.)

The earlier wording confused me because all the data has already been given (the morphism Ω →* is unique since * is terminal, I don’t suggest adding this). I asked myself “what else is there which the property asserts exists” to finish specifying the diagram.

This is my first comment, I hope it’s helpful. Seth

• CommentRowNumber18.
• CommentAuthorUrs
• CommentTimeAug 7th 2022

Okay, thanks. I have now (here) adjusted the wording and beautified the typesetting a little.

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeJan 5th 2023

cross-links with boolean domain

• CommentRowNumber20.
• CommentAuthorUrs
• CommentTimeJan 31st 2023

fixing the pdf link for

• William Lawvere, Quantifiers and sheaves, Actes Congrès intern. Math. 1 (1970) 329-334 [pdf]

here and elsewhere

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