I am reading the appendix of Charles Weibel's Homological Algebra and have the following question. It is mentioned that every morphism $f: B \to C $ in an abelian category factors as $B \to im(f) \to C$ where $im(f)\equiv ker(coker(f))$ and the morphism $B \to im(f)$ is epi and $im(f) \to C$ is mono. I am able to prove the other parts of the problem but not able to show that the morphism $B \to im(f)$ is epi. Please help. Thanks.

  • $\begingroup$ This appendix is too short to learn the basics properly. Better consult Mac Lane's CWM, the chapter on abelian categories. $\endgroup$ Jan 13 '14 at 21:21
  • 1
    $\begingroup$ What is the definition of Abelian Category in that book? $\endgroup$
    – Berci
    Jan 13 '14 at 23:44
  • $\begingroup$ A link where it is shown that the factorization is "universal" and that if the category has equalizers and mono are regular then it factorises as mono $\circ $ epi. (The proofs are in the two last paragraph, not right after the statements are made) $\endgroup$
    – Noix07
    Oct 19 '18 at 13:40

A crucial property of Abelian categories (mostly part of the definition) is that the canonically arising morphism ${\rm coim}(f)\to {\rm im}(f)$ is iso for all morphisms $f$.

Using this, you're done, because, by dual argument you have $B\to{\rm coim}(f)$ is epi.

From the assumptions that every mono is kernel and every epi is cokernel, one way to prove $v:B\to{\rm im}(f)$ is epi (or, equivalently, $u:{\rm coim}(f)\to{\rm im}(f)$ is epi) is as follows: Suppose $t\circ v= 0$ and consider the pushout of ${\rm im}(f)\to C$ and $\,t$:

$B \overset{f}\longrightarrow C \to {\rm coker}(f) \\ v\searrow \, \nearrow \ \searrow t_1 \\ \ \ \ \, {\rm im}(f)\quad\ \ \ \cdot \\ \ \ \ \ \ \ t \searrow \ \nearrow i_1 $

Here $t_1\circ f=0$, so $t_1$ goes through ${\rm coker}(f)$, but then $i_1\circ t$ also becomes $0$.

Now the following lemma ensures that $i_1$ is mono, so $t=0$.

Dually one can prove that ${\rm coim}(f)\to{\rm im}(f)$ is also mono, thus, is a kernel and epi, hence iso.

Lemma: In an Abelian category the pushout of a monomorphism is monomorphism, i.e. if $i:A\hookrightarrow B$ and $f:A\to C$ then the arising arrow $i_1:C\to P$ in the pushout diagram is mono.

$\ \phantom{f} A\overset{i}\hookrightarrow\, B \\ f\downarrow\ \ \ \ \ \downarrow \\ \ \phantom{f} C \underset{i_1}\to P $

Hint: Write the pushout as $P={\rm coker}\left(A\overset{[i,-f]}\longrightarrow B\oplus C\right)$ and observe that $[i,-f]:A\to B\oplus C$ is mono (because of $i$), so you can use the condition that $[i,-f]=\ker(B\oplus C\to P)$.

  • $\begingroup$ Thank you. But I should have clarified the definition of abelian category in use. It is an additive category such that (i) every map has a kernel and a cokernel (ii) a monic is the kernel of its cokernel (iii) every epi is the cokernel of its kernel. Could you please add some comment in the view of this definition ? $\endgroup$
    – user90041
    Jan 14 '14 at 3:39
  • $\begingroup$ I have updated my answer. $\endgroup$
    – Berci
    Jan 15 '14 at 1:46
  • $\begingroup$ Thanks a lot ! I was stuck on it as I did not know what a push-out was. I read the Wikipedia article on that and can understand your answer now. :-) One more request : Could you please suggest me some good place to learn Abelian categories ? Weibel's appendix is too compact. $\endgroup$
    – user90041
    Jan 16 '14 at 5:42
  • $\begingroup$ MacLane's Category Theory for the Working Mathematician, for example... And, if you didn't know the pullback, you could have defined $P$ as ${\rm coker}\left([i,-f]\right)$. $\endgroup$
    – Berci
    Jan 18 '14 at 1:36

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