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In category of abelian groups, we know that

— values of $\rm{lim}^1$ on countable systems are precisely cotorsion groups

— values of $\rm{lim}^1$ on systems of finitely generated groups are of the form $\rm{Ext}^1(A, \Bbb Z)$ with $A$ flat

— every group is a $\rm{lim}^1$ of a system of cardinality $\Omega$

Can these be generalized in some way to an arbitrary abelian category? (Probably, of finite global dimension, otherwise there's little hope for anything similar)

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    $\begingroup$ Isn't the theory of cotorsion pairs a vast generalization of these cases? $\endgroup$
    – David White
    Commented Apr 12, 2019 at 21:20
  • $\begingroup$ Maybe: I do not see how one can prove either second or third statement not using an explicit classification of divisible abelian groups. If you can, I'll be pleased to know how. $\endgroup$
    – Denis T
    Commented Apr 12, 2019 at 23:56

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