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It is known that in the category of sets the dualization of the notion of a subobject classifiers does not work because the only object admitting a morphism into an initial object is the empty set.

But if we look at the idea of a subobject classifier (to index subobjects), then we can see that in the category of sets quotient objects (defined as equivalence classes of epimorphisms that start from a given object) have canonical representatives: those are equivalence relations on the given set.

So, in the category of sets quotient objects of an object $a$ can be classified by the set $E(a)\subseteq \mathcal P(a\times a)$ of all equivalence relations on $a$.

Question. Has the set $E(a)$ of equivalence relations some categorial counterpart, which would index all quotient objects of a given object?

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  • $\begingroup$ What you've written about sets works just as well in any topos. The equivalence relation $\sim$ associated to an epimorphism $f:a\to b$ can be described as the equalizer of the two maps obtained by composing $f$ with the two projections $a\times a\to a$. In fact, this construction of $\sim$ and the fact that $f$ can be recovered as the coequalizer of the projections restricted to $\sim$ needs much less than a topos; a regular category should suffice. Of course to then talk about $\sim$ being an "element" of $\mathcal P(a\times a)$ requires $\mathcal P$ and thus essentially a topos. $\endgroup$
    – Andreas Blass
    Commented May 25, 2020 at 17:13
  • $\begingroup$ @AndreasBlass It is a bit strange that the existence of a quotient object classifier requires the existence of a subobject classifier. Then the general philosophy whould imply that the existence of a subobject classifier is equivalent to the existence of a (properly defined) quotient object classifier, which is a bit strange. $\endgroup$
    – Taras Banakh
    Commented May 25, 2020 at 17:25
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    $\begingroup$ Check out mathoverflow.net/questions/7776/… $\endgroup$
    – Steven Gubkin
    Commented May 25, 2020 at 17:57
  • $\begingroup$ @StevenGubkin Thank you for the link. Indeed, very close question. I tried to find something relatined on MO before writing this question but without success. $\endgroup$
    – Taras Banakh
    Commented May 25, 2020 at 18:17
  • $\begingroup$ Even in a topos, there's only a partial parallel between subobjects and quotients. For any $a$, one has an "object of subobjects" $\mathcal P(a)$ and an "object of quotients" in the sense of an object of equivalence relations on $a$. One also has a universal subobject, $1\to\Omega$, of which all subobjects (of any $a$) are pullbacks. But there is no universal quotient, of which all quotients are pushouts. This seems (to me) to be an instance of a rather general set-theoretic phenomenon: Quotients are not as nice as subsets. (See next comment.) $\endgroup$
    – Andreas Blass
    Commented May 25, 2020 at 19:41

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One way to write the universal property of this object $E(a)$ is as follows:

a map $x \to E(a)$ is the same as an isomorphism class of epimorphism $x \times a \twoheadrightarrow k$ in $Set/x$, that is a diagram

$$ x \times a \twoheadrightarrow k \to x$$

whose composite is the first projection.

So it does "feel like" a subobject classifier, but it is not the dual notion (the dual of a subobject classifer, would have a universal property specifying what are morphisms out of it, not into it).

For this universal property to make sense in a category $C$, you need pullback of epimorphism to exists, or if you restrict to some specific class of epimorphism, that the class of epimorphism under consideration is stable under pullback.

You can also consider the universal property, equivalent in set, but that might be different in more general category:

a map $x \to E(a)$ is the same an an (isomorphism class of) equivalence relation on $x \times a$ which is included in $\Delta_x \times a \times a$.

Where "equivalence relation" on $z$ means subobject of $z \times z$ satisfying the usual stability properties...

This version of the universal property makes sense more generally in any category with finite limits (pullback of monomorphism is enough actually). The two are equivalent in an exact category.

As pointed out by Andreas Blass in the comment, in an elementary topos, these two universal properties make sense and defines a universal objects.

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  • $\begingroup$ I have just noticed that in your (two) answers you replace epimorphisms (as isomorphic classes) by two other isomorphism classes. But the question was to avoid using isomorphism classes at all, because in NBG formalization of category theory, it is not allowed to form classes of classes. So, whereas subobject classifiers allow us to define an object indexing all subobjects, I do not see from your answer which object can index quotient objects. Maybe in topoi you can use the subobject classifier to do this job by analogy as it is done in the category of set. But what to do in general case? $\endgroup$
    – Taras Banakh
    Commented May 26, 2020 at 17:16
  • $\begingroup$ @TarasBanakh : I'm affraid I don't understand your point. To me a "subobject classifier" is an object $\Omega$ such that $Hom(X,\Omega)$ identifies (functorially) with the class of isomorphisms class of monomorphisms $A \to X$. The fact that the framework you chose to work in cannot in general consider class of class in general doesn't means that on specific instances you cannot constructs a set isomorphic to it. $\endgroup$
    – Simon Henry
    Commented May 26, 2020 at 17:26
  • $\begingroup$ These are justs universal property, you can always rephrase them without mentioning these isomorphism class. for example an "epimorphism classifier E(a)" in the first part of my answer is an object coming with an epimorphism $E(a) \times a \twoheadrightarrow k \rightarrow E(a)$ such that for every diagram $X \times a \twoheadrightarrow p \rightarrow X$, there is a unique map $X \to E(a)$ such that the second diagram is obtained as a pull back of diagram above. This is exactly the same thing, but it no longer involves classes in anyway. $\endgroup$
    – Simon Henry
    Commented May 26, 2020 at 17:29
  • $\begingroup$ Thank you for the explanation. Then this is indeed what I expected: some existing object indexing quotient objects (which are equivalence classes). $\endgroup$
    – Taras Banakh
    Commented May 26, 2020 at 18:31

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