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$\textbf{Short exact sequence}$

The most common type of exact sequence is the '''short exact sequence'''. This is an exact sequence of the form $A \;\overset{f}{\hookrightarrow}\; B \;\overset{g}{\twoheadrightarrow}\; C$ where $f$ is a monomorphism and $g$ is an epimorphism. In this case, $A$ is a subobject of $B$, and the corresponding quotient is isomorphic to $C$: $C \cong B/f(A)$ (where $f(A)$ = im($f$)).

Now, does this make sense to say that $A$ is a subobject of $B$ rather than $f(A)$ is a subobject of $B$? My understanding of category theory is that you can play madlibs with it and get a coherent statement in other environments.

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A subobject is an equivalence class of monomorphisms (definition). Hence $A \to B$ represents a subobject. – Martin Brandenburg Dec 8 '13 at 18:16
up vote 4 down vote accepted

If we're talking about modules, for example, then I would say it's fine to call $A$ a subobject of $B$. As $f$ is injective $A$ is isomorphic to $f(A)$ and isomorphisms are commonly repressed when speaking about algebraic objects.

On the other hand, if we're being completely abstract then a subobject of $B$ is an equivalence class of monomorphisms into $B$. In this case neither $A$ nor $f(A)$ is technically a subobject of $B$, the maps $A \to B$ and $f(A) \to B$ are representative elements for a subobject, and in this case they represent the same subobject.

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If we're being completely abstract, $f(A)$ isn't even define a priori. – Pece Dec 8 '13 at 18:11
Well, the image of $f$ is defined and in an abstract setting I would take $f(A)$ to be notation for that. – Jim Dec 8 '13 at 19:55
How do you define the image of $f$ in full generality ? The only notion of an image of an arrow I am aware about is in categories with factorization epi-mono of each arrow (like in a topos for example). – Pece Dec 9 '13 at 7:03
You have such a factorization in abelian categories which is a standard setting for talking about exact sequences. (I presume the OP's not asking about Quillen exact categories) – Jim Dec 9 '13 at 18:51
Ok. As your post were talking about subobject independently of the context of the OP, I assumed you were in full generality… My bad. – Pece Dec 9 '13 at 20:16

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