Is a left-exact limit-preserving functor $Ab \to Ab$ necessarily representable?

Let $Ab$ be the category of abelian groups, and let $F: Ab \to Ab$ be a covariant functor which is left-exact and limit-preserving. Is $F$ necessarily naturally equivalent to a functor of the form $\mathrm{Hom}(A,-)$ for some $A\in Ab$?

• If $F$ also preserves $\kappa$-filtered colimits for some cardinal $\kappa$ the answer is `yes'. In this case, $F$ admits a left adjoint $G$ and the object in question is $G(\mathbb{Z})$. – Dylan Wilson Mar 23 '16 at 18:51
• and since every corepresentable functor has this property, your question can be translated into: "Do there exist limit preserving functors from Ab to Ab which don't preserve filtered colimits?" seems like the answer should be yes but I'll have to think of a counterexample... – Dylan Wilson Mar 23 '16 at 18:53
• I guess it's not so obvious since the result is maybe true for Set... – Dylan Wilson Mar 23 '16 at 18:57
• Limit-preserving implies left exact. – Qiaochu Yuan Mar 23 '16 at 18:57
• @QiaochuYuan As can be seen from the answer below, this is closely related to the (non)existence of a cogenerator. Arbitrarily large simple groups are used in the proof that nonabelian groups do not have a cogenerator (just looked up this proof (1.64 on p.104) in Manes' "Algebraic Theories"). And the same are used to construct a non-representable limit-preserving functor in the nlab link of yours. – მამუკა ჯიბლაძე Mar 23 '16 at 19:43

The category of abelian groups is small-complete, well-powered, and has a cogenerator (e.g., $\mathbb{Q}/\mathbb{Z}$). It follows from the Special Adjoint Functor Theorem that any limit-preserving functor $G: Ab \to Ab$ has a left adjoint $F$. (A proof of the SAFT may be found on this nLab page.) And as Dylan Wilson pointed out in a comment, we then have $G \cong \text{Hom}(\mathbb{Z}, G-) \cong \text{Hom}(F(\mathbb{Z}), -)$, so $G$ is representable.
Referring to another comment by Dylan: all such functors are necessarily accessible, since any abelian group is $\kappa$-presentable for some $\kappa$.
• @DylanWilson Cototality was actually the first word that popped into my mind when I read the question; it was just a matter of recalling why $Ab$ was cototal. As you may know, a functor from a cototal category to a locally small category has a left adjoint iff it preserves all small limits. Locally presentable categories need not be cototal (as witnessed by $Grp$, as Qiaochu was saying), but any accessible limit-preserving functor between locally presentable categories has a left adjoint. Is this responsive to your question (not sure I parsed it right)? – Todd Trimble Mar 24 '16 at 0:30
• Note that it is a particular case of a theorem from homological algebra known as Watts theorem: every limit-preserving functor from the category of left $R$-modules to abelian groups is representable. – Philippe Gaucher Mar 24 '16 at 9:14