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Can you give such example? If not, maybe you can give any reason why there are such abelian categories?

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Abelian groups objects on a Topos isnt necessarly well powred. Is interesting make from topoi abelian category as abelian groups or modules as generaization of sheaves categories, (sheaves on Grothendieck sites) – Buschi Sergio Apr 12 '12 at 11:30
Are there more easy examples? – Martin Brandenburg Apr 12 '12 at 14:04

In Appendix C (Corollary C.3.3 to be precise) of Neeman's book "Triangulated Categories" an example of an abelian category which is not well-powered is given.

The actual counterexample is given by $A(D(R))$ where $D(R)$ is the unbounded derived category of a discrete valuation ring $R$, and $A(D(R))$ is the category of finitely presented additive functors $D(R)^\mathrm{op} \to Ab$.

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Is this category locally small? – Anonymous Apr 12 '12 at 12:58
Yes, this follows from the fact that one is only looking at the finitely presented functors together with Yoneda. – Greg Stevenson Apr 12 '12 at 13:18

There is a construction of Peter Freyd that embeds any triangulated category $\mathcal{T}$ in an abelian category $\mathcal{A}(\mathcal{T})$. Explicitly, we start with the category of arrows in $\mathcal{T}$. Given a morphism $u$ in $\mathcal{T}$, I'll write $I(u)$ for the same thing regarded as an object of the arrow category. Next, we identify two morphisms $(f,g):I(u)\to I(v)$ and $(f',g'):I(u)\to I(v)$ if the diagonal composite $vf=gu$ is the same as the diagonal composite $vf'=g'u$. It is straightforward to see that this gives a quotient category, which we call $\mathcal{A}(\mathcal{T})$. It is quite a long and interesting argument to show that this is actually an abelian category. It is almost never well-powered unless $\mathcal{T}$ is small.

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