Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
add comment

1 Answer

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 $X_i$.

share|improve this answer
    
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.