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.



are quotients by equivalence relations "better" than surjections?
It seems that you've got factorization of maps covered, so let me address the question of why canonical quotient maps and canonical inclusions are "better".


Given a set $X$, in general there is a proper class of injections $Y \to X$. However, many of these are isomorphic, where injections $i : Y \to X$ and $j : Z \to X$ are isomorphic if there is an isomorphism $k : Y \to Z$ such that $i = j \circ k$. The isomorphism classes of injections into $X$ are the *subobjects* of $X$. In fact, there are only set-many subobjects of $X$ (in category-theoretic language, sets form a [well-powered category](https://ncatlab.org/nlab/show/well-powered+category)). It is pesky to work with set-many proper equivalence classes, so we instead look for a *set* $P(X)$ of injections into $X$, one from each isomorphism class. We may additionally require some nice properties, for instance, if $i : X \to Y$ is in $P(Y)$ and $j : Y \to Z$ is in $P(Y)$, we would expect $j \circ i : X \to Z$ to be in $P(Z)$. One can come up with a wish list of such nice closure conditions, here's another one: if $i : Y \to X$ and $j : Z \to X$ are in $P(X)$, and there is (a unique) $k : X \to Z$ such that $i = j \circ k$, then $k$ is in $P(Z)$.


We know the answer, of course, just take $P(X)$ to be the canonical subset inclusions into $X$. This is not the only choice of such representative inclusions, but it's a pretty good one.


We may therefore say that the canonical inclusions of subsets are "better" because they are the canonical representatives of subobjects (equivalence classes of injections).


The answer for quotient maps and surjections is dual. Consider equivalence classes of surjections, quotiented by isomorphism. There are only set-many such classes, therefore sets form a *well-copowered category*. (Some people say "cowell-powered" but then why not call it "ill-powered"?) This time we look for a set $Q(X)$ of surjections from $X$, each representing one equivalence class of surjections from $X$. We may take $Q(X)$ to be the set of all canonical quotient maps $X \to X/{\sim}$, or just the set of all equivalence relations on $X$. Once again, canonical quotient maps are "better" because they are the distinguished representatives of isomorphism classes of surjections.


