$\require{AMScd} \newcommand{\cat}{\mathcal{C}} \newcommand{\coker}{\mathrm{coker}} $The following definitions are from Tensor Categories by Etingof et al. This is all set in abelian categories.
A monic (epic) is a morphism with trivial kernel (cokernel)
I know that this is equivalent to the "usual" $f\circ g_1=f\circ g_2\implies g_1=g_2$, but I want to try to use only this definition.
A subobject of $X$ is an object $Y$ with a monic $i:Y\rightarrow X$
A quotient of $X$ is an object $Y$ with an epic $p:X\rightarrow Y$
A subquotient of $X$ is a quotient of a subobject
Now, I want solve Exercise $1.3.6$:
Show that a subquotient of a subquotient of $Y$ (in particular, a subobject of a quotient of $Y$) is a subquotient of $Y$
So the situation is as follows:
$\begin{CD} X_1\\ @AAi_2A\\ X_2 @>p_2>> X_3\\ @. @AAi_4A\\ @. X_4 @>p_4>>X_5 \end{CD}$
I hope the reason I labelled the arrows the way I did is obvious. To prove the given statement, I think we must find (show the existence of) an object $Y$ together with
- a monomorphism $Y\xrightarrow{i_Y}X_1$ (well, $X_2$ would suffice)
- an epimorphism $Y\xrightarrow{p_Y}X_4$ (then, $p_4\circ p_Y$ is epic)
But how? I have tried numerous things. I have found that $\text{cokernel(monic)}=\text{epic}$ and its dual - looking it up, this seems to coincide in a way with the notion of regular, is that correct?
Unfortunately, this didn't give me any new insight. I tried looking for some universal property because of the definition via isomorphism classes on the nLab and other posts on this site, but that was probably misguided?
Just hints would me much appreciated, a quick nudge in the right direction perhaps.
Thanks!
The usual way to "compose" two spans to get a third span is to fill in the diagram with a fourth span. (I hope you don't mind if I change the orientation to the usual picture)
$$\require{AMScd} \begin{CD} Y @>>> X_4 @> p_2 >> X_5 \\ @VVV @VV i_2 V \\ X_2 @> p_1>> X_3 \\ @VV i_1 V \\ X_1 \end{CD} $$
Stop reading here if you want to figure the rest out on your own.
In a nice category, the canonical choice of span is the one that makes the square a pullback square.
In any category, pullbacks (when they exist) of monics are monic. Abelian categories have the additional property that pullbacks of epics are also epic.
(usually the dual is stated: pushouts of monics are monic, but abelian categories are self-dual)