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Can you give an example of an abelian category which is not well-powered? If not, maybe you can give any reason why there are such abelian categories?

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  • $\begingroup$ 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) $\endgroup$ Apr 12, 2012 at 11:30
  • $\begingroup$ Are there more easy examples? $\endgroup$ Apr 12, 2012 at 14:04

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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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    $\begingroup$ Is this category locally small? $\endgroup$
    – Anonymous
    Apr 12, 2012 at 12:58
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    $\begingroup$ Yes, this follows from the fact that one is only looking at the finitely presented functors together with Yoneda. $\endgroup$ Apr 12, 2012 at 13:18
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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.

This is from the "Proceedings of the Conference on Categorical Algebra" held in La Jolla in 1965, published by Springer.

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  • $\begingroup$ Can you give a reference for this construction? $\endgroup$ Nov 29, 2017 at 13:39
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    $\begingroup$ @IvanDiLiberti I have added a reference. $\endgroup$ Nov 29, 2017 at 13:46
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Here's a simpler, but less consequential, example.

Take the category of "eventually constant" functors from ordinals (considered as a category with a single morphism $\alpha\to\beta$ when $\alpha\leq\beta$) to abelian groups, meaning functors $F$ for which there is some ordinal $\alpha$ such that $F(\beta)\to F(\gamma)$ is an isomorphism for all $\alpha\leq\beta\leq\gamma$.

The "eventually constant" condition ensures that this is a locally small category. It is not well-powered, since for any ordinal $\alpha$, the constant functor taking value $\mathbb{Z}$ has a subfunctor $F_\alpha$ with $$F_\alpha(\beta)=\begin{cases}0&\text{if $\beta<\alpha$}\\ \mathbb{Z}&\text{if $\beta\geq\alpha$.} \end{cases}$$

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