TITLE: are quotients by equivalence relations "better" than surjections?
QUESTION [22 upvotes]: This might be a load of old nonsense.
I have always had it in my head that if $f:X\to Y$ is an injection, then $f$ has some sort of "canonical factorization" as a bijection $X\to f(X)$ followed by an inclusion $f(X)\subseteq Y$. Similarly if $g:X\to Y$ is a surjection, and if we define an equivalence relation on $X$ by $a\sim b\iff g(a)=g(b)$ and let $Q$ be the set of equivalence classes, then $g$ has a "canonical factorization" as a quotient $X\to Q$ followed by a bijection $Q\to B$. Furthermore I'd always suspected that these two "canonical" factorizations were in some way dual to each other.
I mentioned this in passing to a room full of smart undergraduates today and one of them called me up on it afterwards, and I realised that I could not attach any real meaning to what I've just said above. I half-wondered whether subobject classifiers might have something to do with it but having looked up the definition I am not so sure that they help at all.
Are inclusions in some way better than arbitrary injections? (in my mind they've always been the "best kind of injections" somehow). Are maps to sets of equivalence classes somehow better than arbitrary quotients? I can't help thinking that there might be something in these ideas but I am not sure I have the language to express it. Maybe I'm just wrong, or maybe there's some ncatlab page somewhere which will explain to me what I'm trying to formalise here.

REPLY [4 votes]: "Are inclusions in some way better than arbitrary injections?"

No they are not.
Anything that you can say about the one, you can say about the other.
In particular:
Given $X$, there is a set of inclusions into $X$. And there is a (large groupoid equivalent to a) set of injections into $X$. For all practical purposes these are the same thing. So they should be treated as the same thing.

Are quotient maps somehow better than arbitrary surjections?

Again no.
Same story.
    The fact that we can talk about inclusions versus injections is an artefact of our set theoretic foundations of mathematics. It would be better to use a language that disallows us from asking whether two sets are equal, and only allows us to talk about isomorphisms between sets.    We largely already do this in practice. When we write $\mathbb Z \subset \mathbb Q$, we don't stop to explain that elements of $\mathbb Q$ are defined as pairs of elements of $\mathbb Z$, and we don't distinguish between $\mathbb Z$ and its image in $\mathbb Q$. We just write $\mathbb Z \subset \mathbb Q$.
    Unfortunately, I'm not knowledgeable enough in foundational issues in order to make sense of the notion of "a language that disallows asking whether two sets are equal, and only allows to talk about isomorphisms". But I'm sure that such a thing exists (and that it can be tweaked so as to be exactly as powerful as ZFC).