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The standard categorical definition of image is that it is the cokernel of the kernel. Under what nice conditions does this definition coincide with kernel of the cokernel? It coincides for abelian groups, modules,... and I'd dare to say that over any category of sets with "some additional structure" as it is usually vaguely defined in category books.

Rodrigo
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1 Answers1

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An additive category with kernels and cokernels is an abelian category iff the canonical morphism $\mathrm{coker}(\ker(f)) \to \ker(\mathrm{coker}(f))$ is an isomorphism for all morphisms $f$. See here for a list of non-examples. For example, the category of Banach spaces with continuous linear maps is an additive category with kernels and cokernels (mod out the closure of the image), but not abelian. The morphism above is for $f : V \to W$ the natural morphism $V/\ker(f) \to \overline{\mathrm{im}(f)}$, which is an isomorphism iff $\mathrm{im}(f)$ is closed.

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Martin Brandenburg
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    Thank you! As I looked more closely into abelian cat's, I found that in prop 9.6 Hilton & Stammbach proved that in an abelian category any morphism $\phi : A \rightarrow B$ factors as $K \hookrightarrow A \twoheadrightarrow I \hookrightarrow B \twoheadrightarrow C$. This factorization is equivalent to your isomorphism $\def\coker{\text{coker}}\coker (\ker \phi) \xrightarrow{\sim} \ker (\coker\phi)$. Conversely, if it's true for every morphism, then clearly we have that (co)kernels are (co)normal, and that every morphism factors as an epi composed with a mono, so we are in an abelian cat. – Rodrigo Oct 27 '13 at 19:10