''Labelling discrimination'' for objects in a category

I am not particularly used to the category theory thinking paradigm and there are certain statements that I am used to making in the set theoretic modelling approach that I would like to characterize in the categorical language. I would simply like feedback from more experimented categoricians on a possible solution to my problem that I have exposed at the end of my comment.

In the context of Sets, when I asked (my supervisor) about how one could express the idea of a subset or of intersection of sets in the category framework, I was refered to equalizers and pullbacks. And of course, the point of working in a category is that up to isomorphism, you can't distinguish objects by their arrows. If we defined ''having an equalizer into ...'' as ''being a subset of ...',' then given a subset X of Y, and any set Z that is isomorphic to X, we would be able to find an equalizer from Z into Y and that would make Z a subset of Y. And that's fine, in the abstract sense.

In the usual set theoretic approach, the labelling of elements allows me to discriminate amongst say, X={1,2} and Z={1,4}, even though they are isomorphic to each other as sets. Now, I can say that X is a subset of Y={1,2,3}, but Z is not a subset of the latter.

The point I'm getting to is that I would like to make the above kind of distinction amongst objects, yet remain as much as possible in the category theory framework. Ideally, I would be able to see this property of having ''actual'' subsets of a set (carried out by finding those sets that have sublabellings of the latter) as an abstract external property on sets as objects of a category (that is, without the need to provide explicit internal labels).

MY QUESTION IS THIS : What kind of structure could I add on top of a category that would allow me to recover the possiblity of distinguishing between subobjects of an object Y that are ''actual'' subobjects of Y as opposed to those that aren't (as in the above example, where X is a subset of Y, but not Z).

My guess is that, given a category C, all I require is a partial order R on objects, that describes the ''actual subobject relations''. Thus, related objects with R(A,B) are required to have at least an equalizer from A into B. Yet, the existence of an equalizer from some A into some B does not necessarily imply R(A,B) (so that we may have copies of A lying around but not necessarily directly subobjects of B).

To put it in a purely categorical language, I think it would be the equivalent of saying that there exists a functor from some poset into the category C in question, that maps morphisms of the poset to equalizers of C. Does that make any sense or is it complete dissonance from my part and misuse of the categorical framework? Also, I might be missing a condition on this ''functor'' that would render the idea correctly, I'm not sure.

I would appreciate any feedback and/or references on the matter.

Thank you

What you view as the "actual subobject relation" is a notion of set theory which has no meaning in category theory. Actually, in my opinion, category theory offers the "right" perspective on subobjects (subsets, subgroups, subrings, subspaces, $\dotsc$). Namely, that "being a subobject" is not a relation on the class of objects, but rather a class of morphisms! This way we remember how two objects embed into each other, and not just that they embed. Details:

A monomorphism in a category is a morphism $f : B \to A$ which has the property that $fx=fy \Rightarrow x=y$ for all morphisms $x,y : T \to B$. In many categories in everyday mathematics, the monomorphisms are exactly what you expect them to be.

A subobject of an object $A$ is a monomorphism $f : B \to A$. The object $B$ can be recovered as the domain of $f$ here. But notice that $f$ cannot be recovered from $B$. It is really $f$ that is the subobject of $A$, not $B$!

For example: A subset of a set $A$ is an injective map $B \to A$. A subgroup of a group $A$ is an injective group homomorphism $B \to A$. A subspace of a topological space $A$ is an injective continuous map $B \to A$.

In practice one often says that $B$ is a subobject of $A$ when there is a canonical $f : B \to A$ in the context and one actually means that $f$ is a subobject. For example when we say that $\mathbb{Z}$ is a subset of $\mathbb{Q}$, we actually mean that $z \mapsto \frac{z}{1}$ is an injective map!

We can compare subobjects: If $f: B \to C,g : B' \to C$ are subobjects of $A$, we write $f \subseteq g$ when there is a morphism $i : B \to B'$ with $g \circ i = f$. In particular, we write $f \cong g$ and call $f,g$ isomorphic or equivalent when $f \subseteq g \subseteq f$, which means that there is an isomorphism $i : B \to B'$ with $g \circ i = f$.

(In many texts you will see that isomorphic subobjects are identified to be the same, but this is a) not really necessary to work with subobjects, b) not natural from the perspective of higher category theory.)

Now let us apply this to the category of sets and the examples you have mentioned (probably many set theorists won't agree with me here):

$B=\{1,2\}$, $A=\{1,2,3\}$, $C=\{1,4\}$

Let us denote by $i_A : A \to \mathbb{N}$ the inclusion map, likewise for $i_B$ and $i_C$.

According to the definition above, it doesn't make sense to say that $B$ is a subset of $A$. But the inclusion map $f : B \to A$ makes $f$ a subset of $A$. But what we actually mean is that the subsets $(B,i_B)$ and $(A,i_A)$ of $\mathbb{N}$ satisfy $(B,i_B) \subseteq (A,i_A)$. When we write $B \subseteq A$, we actually mean this.

It is correct that $B$ and $C$ are isomorphic, and any isomorphism $B \to C$ can be regarded as a (terminal) subset of $C$. But $(B,i_B)$ and $(C,i_C)$ are not isomorphic subsets of $\mathbb{N}$. And this is what we actually mean by $B \not\subseteq C$!

Finally, there is a monomorphism $f : C \to A$, for example $f(1)=1$ and $f(4)=2$, which is then a subset of $A$. But still $(C,i_C) \not\subseteq (A,i_A)$, what we actually mean by $C \not\subseteq A$.

When doing set theory, we often think of all sets (in a given context) as part of a large set, the universe $V$. When $A,B$ are sets in $V$, then what set theorists write as $B \subseteq A$, is not the statement that $B$ is a subset of $A$ in the sense of category theory, but rather that $(A,i_A) \subseteq (B,i_B)$ where $i_A : A \to V$ and $i_B : B \to V$ are the inclusions. In other words, set theorists don't remember the inclusion to the universe. I think they should! Besides, there is just no set $V$ containing all other sets, so that one has to choose $V$ in "a higher world" (there are several ways to make this precise) and it doesn't belong to the given category of sets anymore. Another reason against this idea of a big universe is that it just doesn't exist for other categories (of course we can formally adjoin a terminal object to any given category, but this has no actual meaning).

A quite related question is the following: What does it mean for two sets $A,B$ to be disjoint? Of course ZF or any classical axiomatization of set theory offers an answer. But does this really make sense? Are $\pi+5$ and $\ln(2)$ disjoint? (Remember that in ZF everything is a set, even numbers.) For a category theorist it doesn't make sense to ask if two sets $A,B$ are disjoint, since this relation is not invariant under equivalences of categories $\mathsf{Set} \to \mathsf{Set}$. It makes much more sense to embed $A,B$ into a given large set $C$ and then ask if $A$ and $B$ are disjoint in $C$. More generally, if $f : A \to C$ and $g : B \to C$ are subobjects (or arbitrary morphisms) in a category with pullbacks, then we say that $f$ and $g$ are disjoint when the pullback $A \times_C B$ is an initial object.

I hope that I could convince some of the readers that category theory offers the "correct" perspective on subobjects.

While I agree with the sentiment in the other answer, I feel compelled to offer a counterpoint.

In Categories, Allegories, by Freyd, Scedrov, there is a section on what they call $\tau$-categories. I don't have a copy of the book handy and I don't remember the details for sure, but I recall it being a structure that amounts to distinguishing specific limit diagrams (e.g. pullbacks, products, equalizers, terminators); e.g. for a given pair of objects, there is a unique product and projection maps that form a distinguished diagram.

The $\tau$ structure may allow you to do the sorts of things you were seeking to do in your post.