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Suppose in some nice enough category, say abelian groups, we have a filtered colimit of compact objects, might as well say a colimit indexed by $\mathbb{N}$. If we are given an element $x$ in the colimit, must it have some representative at a finite stage? If so does this follow from the colimit being filtered, from the objects being compact, or both?

Thanks! Jon

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up vote 1 down vote accepted

Compactness and filtered are two different notions -though related. But the point is not the colimit being of compact objects.

The condition of being the colimit filtered already implies that any element $x \in \mathrm{colim}_i X_i$ has a representative in some $X_i$. No compact objects needed here.

The compactness condition for an object $K$ means that, for a filtered colimit, the universal morphism

$$ \mathrm{colim}_i \mathrm{Hom} (K, X_i) \longrightarrow \mathrm{Hom} (K, \mathrm{colim}_i X_i ) $$

is a bijection. In particular, any morphism $K \longrightarrow \mathrm{colim}_i X_i$ factors through some $X_i$.

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Ah yes, thankyou! That second statement was one I had seen before, but could not remember the exact phrasing. – Jon Beardsley Oct 13 '11 at 18:06

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