Abelian category which is not well-powered

Question

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?

Answer 1

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 $D(R)^\mathrm{op} \to Ab$.

Answer 2

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.

Answer 3

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 $$F_\alpha(\beta)=\begin{cases}0&\text{if $\beta<\alpha$}\\ \mathbb{Z}&\text{if $\beta\geq\alpha$.} \end{cases}$$