Mixing
Perfect class of morphisms closed under retracts?
$begingroup$
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...
category-theory model-categories
asked Jan 7 at 15:43
Oscar P.Oscar P.
766
$endgroup$
add a comment |
$begingroup$
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...
category-theory model-categories
asked Jan 7 at 15:43
Oscar P.Oscar P.
766
$endgroup$
add a comment |
$begingroup$
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...
category-theory model-categories
asked Jan 7 at 15:43
Oscar P.Oscar P.
766
$endgroup$
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...
category-theory model-categories
category-theory model-categories
asked Jan 7 at 15:43
Oscar P.Oscar P.
766
asked Jan 7 at 15:43
Oscar P.Oscar P.
766
asked Jan 7 at 15:43
Oscar P.Oscar P.
766
asked Jan 7 at 15:43
Oscar P.Oscar P.
766
asked Jan 7 at 15:43
Oscar P.Oscar P.
766
766
add a comment |
add a comment |
1 Answer
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$begingroup$
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 $(h,k): fto g$ in the arrow category with splitting $(i,j)$, we may write $g$ as the colimit of the filtered diagram $fto fto fto...$, 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.
answered Jan 8 at 20:53
Kevin CarlsonKevin Carlson
32.8k23372
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$begingroup$
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 $(h,k): fto g$ in the arrow category with splitting $(i,j)$, we may write $g$ as the colimit of the filtered diagram $fto fto fto...$, 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.
answered Jan 8 at 20:53
Kevin CarlsonKevin Carlson
32.8k23372
$endgroup$
add a comment |
$begingroup$
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 $(h,k): fto g$ in the arrow category with splitting $(i,j)$, we may write $g$ as the colimit of the filtered diagram $fto fto fto...$, 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.
answered Jan 8 at 20:53
Kevin CarlsonKevin Carlson
32.8k23372
$endgroup$
add a comment |
$begingroup$
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 $(h,k): fto g$ in the arrow category with splitting $(i,j)$, we may write $g$ as the colimit of the filtered diagram $fto fto fto...$, 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.
answered Jan 8 at 20:53
Kevin CarlsonKevin Carlson
32.8k23372
$endgroup$
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 $(h,k): fto g$ in the arrow category with splitting $(i,j)$, we may write $g$ as the colimit of the filtered diagram $fto fto fto...$, 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.
answered Jan 8 at 20:53
Kevin CarlsonKevin Carlson
32.8k23372
answered Jan 8 at 20:53
Kevin CarlsonKevin Carlson
32.8k23372
answered Jan 8 at 20:53
Kevin CarlsonKevin Carlson
32.8k23372
answered Jan 8 at 20:53
Kevin CarlsonKevin Carlson
32.8k23372
32.8k23372
add a comment |
add a comment |
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