Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

$\textbf{Short exact sequence}$

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.

share|improve this question
    
A subobject is an equivalence class of monomorphisms (definition). Hence $A \to B$ represents a subobject. –  Martin Brandenburg Dec 8 '13 at 18:16
add comment

1 Answer

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.

share|improve this answer
1  
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
show 1 more comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.