4
$\begingroup$

Suppose we are working in a presentable class of morphisms $\mathcal{C}$. A class of morphisms $P$ is said to be perfect if :

1) The class $P$ contains all isomorphisms.

2) The class $P$ statisfies the two-out-of-three property.

3) It is close under filtered colimits.

4)There exists a set $P_0 \subseteq P$ such that every morphism in $P$ can be obtained as a filtered colimit of morphisms in $P_0$.

The question is then: is perfect class of morphisms closed under retracts? I've stumbled upon some occurences in which it looks like they are using this fact but I can not manage to produce a proof. I was thinking that using the two first axioms it would be simple to show it but I am stuck...

$\endgroup$
2
$\begingroup$

It doesn't follow from the first two axioms. Instead, you must use that any retract can be written as a filtered colimit. Given a split epimorphism $(h,k): f\to g$ in the arrow category with splitting $(i,j)$, we may write $g$ as the colimit of the filtered diagram $f\to f\to f\to...$, where every map is $(ih,jk)$. This isn't special to the arrow category, and uses none of properties 1,2,4: any category closed under filtered colimits is closed under retractions.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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