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…
Best Answer
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.