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Is this the definition of an exact sequence in an Abelian category: $A\xrightarrow{f}B\xrightarrow{g}C$ is exact at $B$ if $ker(g)=im(f)$?

That what it says at the start of page 7 in Weible.

Why does this comment (Weibel IHA Exercise 1.2.4) say otherwise, can someone provide a reference?

Shean
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  • You have to be careful about what is meant by "image." – Randall Feb 15 '23 at 19:40
  • $im(f)$ is the subobject of $ker(coker(f))$, right? In R-mod this has the usual meaning of image. – Shean Feb 15 '23 at 19:43
  • In $R$-mod, yes, but that is a concrete category. – Randall Feb 15 '23 at 19:46
  • I think what Randall is trying to say is: in a general Abelian category, $\operatorname{ker}(g)$ is given by a monomorphism $X \to B$ for some $X$; and similarly, $\operatorname{im}(f)$ is given by a monomorphism $Y \to B$ for some $Y$. What you actually want to say is that these "are the same subobject", in the sense that there is an isomorphism $X \overset{\sim}{\to} Y$ which makes a commutative diagram when combined with the morphisms to $B$. (And then, the maps to $B$ being monic automatically implies uniqueness of such an isomorphism, etc.) – Daniel Schepler Feb 15 '23 at 20:10
  • So, it might be the case that Weible is (implicitly or explicitly) considering the class of monic morphisms with target $B$ to be equipped with the canonical equivalence relation described above, and then using the notation of equality to refer to that canonical equivalence relation. – Daniel Schepler Feb 15 '23 at 20:13

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