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I am hereby moving an old query-box discussion from abelian category to here. I suggest that to the extent this reached a conclusion, that conclusion should be moved to the Properties-section of the entry
[begin forwarded discussion]
The following discussion is about whether a pre-abelian category in which (epi,mono) is a factorization system is necessarily abelian.
+–{: .query} Mike: In Categories Work, and on Wikipedia, an abelian category is defined to be (in the terms above) a pre-abelian category such that every monic is a kernel and every epi is a cokernel. This implies that (epi, mono) is an orthogonal factorization system, but I don’t see why the converse should hold, as this seems to assert.
Zoran Skoda It is very late night here in Bonn, so check on my reasoning, but I think that the answer is simple. Let $f: A\to B$. The canonical map $\coker(\ker f)\to \ker (\coker f)$ exists as long as we have additive category admitting kernels and cokernels. The arrow from A to coker (ker f) is epi as every cokernel arrow, and the arrow of $\ker(\coker f) \to B$ is mono. Now canonical arrow in between the two is automatically both mono and epi. For all that reasoning I did not yet assume the axiom on uniquely unique factorization. Now assume it and you get that the canonical map must be isomorphism because it is the unique iso between the two decompositions of $f$: one in which you take epi followed by (the composition of) two monics and another in which you have (the composition of) two epis followed by one monic. Right ?
Now do this for $f$ a monic and you get a decomposition into iso iso kernel and for $f$ an epi and you get the cokernel iso iso as required.
Mike: Why is the canonical comparison map mono and epi? It’s late for me too right now, but I think that maybe a counterexample is the “multiplication by 2” map $\mathbb{Z}\to \mathbb{Z}$ in the category of torsion-free abelian groups.
However, if you assume explicitly that that comparison map is always an isomorphism, then I believe it for the reasons that you gave.
Zoran Skoda I do not see this as a counterexample, as this is not a pre-abelian category, you do not have cokernels in this category ? In a pre-abelian category always the canonical map from coker ker to ker coker has its own kernel 0 and cokernel 0.
Mike: Torsion-free abelian groups are reflective in abelian groups, and therefore cocomplete. In particular, they have cokernels, although those cokernels are not computed as in Ab. In particular, the cokernel of $2:\mathbb{Z}\to\mathbb{Z}$ is 0.
Zoran Skoda Yes, I was thinking of this reflection argument (equivalence of torsion and localization argument), that is why I put question mark above. Now I tried to prove the assertion that in preabelian cat the canonical map has kernel 0 and cokernel 0 and I can’t for more than an hour. But that would mean that for example Gelfand-Manin book is wrong – it has the discussion on A4 axiom and it says exactly this. Popescu makes an example of preabelian category where canonical map is not iso, but emphasises in his example that it is bimorphism. On the other hand, later, he says that preabelian category is abelian iff it is balanced and the canonical map is bimorphism, hence he requires it explicitly. Let me think more…
Zoran Skoda I have rewritten in minimalistic way, leaving just what I can prove, and assuming that you are right and Gelfand-Manin book has one wrong statement (that the canonical map in preabelian category is mono and epi). But let us leave the discussion here for some time, maybe we can improve the question of the difference between preabelian with factorization and abelian.
Mike: I refactored the page to make clear what we know and what we don’t, and include some examples. Maybe someone will come along and give us a counterexample or a proof. I wonder what the epimorphisms are in the category of torsion-free abelian groups, and in particular whether it is balanced (since if so, it would be a counterexample).
Mike: Okay, it’s obvious: the epimorphisms in $tfAb$ are the maps whose cokernel (in $Ab$) is torsion. Thus $2:\mathbb{Z}\to\mathbb{Z}$ is monic and epic, so $tfAb$ is not balanced. And since $2:\mathbb{Z}\to\mathbb{Z}$ is its own canonical map, that canonical map is monic and epic in $tfAb$, so this isn’t a counterexample.
Zoran: http://www.uni-trier.de/fileadmin/fb4/INF/TechReports/semi-abelian_categories.pdf says at one place that Palamodov’s version of semi-abelian category is preabelian + canonical morphism is epi and mono. =–
[end forwarded discussion]
I added the (non-)conclusion of the discussion as a remark after the property of having an (epi,mono) factorization system.
Thanks!
I have tried to make the whole discussion in the entry more transparent by organizing it by topics in the Properties-section.
The remarks on regularity and balancing is now at Properties - general. Then what you just edited is now at Properties - factorization of morphisms.
I am going to add further discussion of embedding theorems to abelian category, such as the one by Bergman from his 1974 notes. I am also going to connect to the existing entry element in an abelian category.
But I can’t do anything right now. Since the lab is too busy with something else to serve up pages…
Okay, I have restarted the server. Now it reacts again.
I have turned at abelian category the section on Freyd-Mitchell embedding into a subsection of a more general section Embedding theorems and added a bit more motivating discussion there.
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