Weak factorization systems with right maps non closed under retracts
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Is there a standard name for the following flavor of weak factorization system?
I am interested in the data of two classes of maps $mathfrak L$ and $mathfrak R$ in a category $mathcal C$ such that:
- $mathfrak L$ is exactly the class of element having the left lifting property against all the elements of $mathfrak R$
- $mathfrak R$ is closed under pullback
- each morphism of $mathcal C$ can be factored as an element of $mathfrak L$ postcomposed with a element of $mathfrak R$
So the main difference with actual weak factorization systems is that $mathfrak R$ is not necessarily closed under retracts. I guess it is a notion that is fairly common (or at least the dual version, when dealing with cofibrantly generated model categories), but I have not been able to find a name for this in the litterature.
reference-request category-theory terminology model-categories
asked Aug 27 at 15:56
Pece
8,00711040
add a comment |Â
up vote
1
down vote
favorite
Is there a standard name for the following flavor of weak factorization system?
I am interested in the data of two classes of maps $mathfrak L$ and $mathfrak R$ in a category $mathcal C$ such that:
- $mathfrak L$ is exactly the class of element having the left lifting property against all the elements of $mathfrak R$
- $mathfrak R$ is closed under pullback
- each morphism of $mathcal C$ can be factored as an element of $mathfrak L$ postcomposed with a element of $mathfrak R$
So the main difference with actual weak factorization systems is that $mathfrak R$ is not necessarily closed under retracts. I guess it is a notion that is fairly common (or at least the dual version, when dealing with cofibrantly generated model categories), but I have not been able to find a name for this in the litterature.
reference-request category-theory terminology model-categories
asked Aug 27 at 15:56
Pece
8,00711040
The closest thing I can think of is that elements of $R$ can be called relative cocell complexes. As you say, most weak factorization systems used in practice are of this form, but the retract closure is so useful-for other purposes than construcying the factorizations-that it seems people usually just go ahead and close $R$ up under retracts.
– Kevin Carlson
Aug 27 at 17:29
@KevinCarlson Thanks, that is more or less what I understood also. However, it seems that the notion defined in my post is a little different from the assertion that $(vphantommathfrak R^perpmathfrak R,(vphantommathfrak R^perpmathfrak R)^perp)$ is a weak factorization system, because in the latter the factorization of some $f$ as $pi$ can yield $p$ as only a retract of an element of $mathfrak R$, while the former require $p$ to be actually in $mathfrak R$. Or am I missing something that makes the two notions equivalent?
– Pece
Aug 28 at 9:25
No, you're not missing anything.
– Kevin Carlson
Aug 28 at 16:34
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Is there a standard name for the following flavor of weak factorization system?
I am interested in the data of two classes of maps $mathfrak L$ and $mathfrak R$ in a category $mathcal C$ such that:
- $mathfrak L$ is exactly the class of element having the left lifting property against all the elements of $mathfrak R$
- $mathfrak R$ is closed under pullback
- each morphism of $mathcal C$ can be factored as an element of $mathfrak L$ postcomposed with a element of $mathfrak R$
So the main difference with actual weak factorization systems is that $mathfrak R$ is not necessarily closed under retracts. I guess it is a notion that is fairly common (or at least the dual version, when dealing with cofibrantly generated model categories), but I have not been able to find a name for this in the litterature.
reference-request category-theory terminology model-categories
asked Aug 27 at 15:56
Pece
8,00711040
Is there a standard name for the following flavor of weak factorization system?
I am interested in the data of two classes of maps $mathfrak L$ and $mathfrak R$ in a category $mathcal C$ such that:
- $mathfrak L$ is exactly the class of element having the left lifting property against all the elements of $mathfrak R$
- $mathfrak R$ is closed under pullback
- each morphism of $mathcal C$ can be factored as an element of $mathfrak L$ postcomposed with a element of $mathfrak R$
So the main difference with actual weak factorization systems is that $mathfrak R$ is not necessarily closed under retracts. I guess it is a notion that is fairly common (or at least the dual version, when dealing with cofibrantly generated model categories), but I have not been able to find a name for this in the litterature.
reference-request category-theory terminology model-categories
asked Aug 27 at 15:56
Pece
8,00711040
asked Aug 27 at 15:56
Pece
8,00711040
asked Aug 27 at 15:56
Pece
8,00711040
asked Aug 27 at 15:56
Pece
8,00711040
8,00711040
The closest thing I can think of is that elements of $R$ can be called relative cocell complexes. As you say, most weak factorization systems used in practice are of this form, but the retract closure is so useful-for other purposes than construcying the factorizations-that it seems people usually just go ahead and close $R$ up under retracts.
– Kevin Carlson
Aug 27 at 17:29
@KevinCarlson Thanks, that is more or less what I understood also. However, it seems that the notion defined in my post is a little different from the assertion that $(vphantommathfrak R^perpmathfrak R,(vphantommathfrak R^perpmathfrak R)^perp)$ is a weak factorization system, because in the latter the factorization of some $f$ as $pi$ can yield $p$ as only a retract of an element of $mathfrak R$, while the former require $p$ to be actually in $mathfrak R$. Or am I missing something that makes the two notions equivalent?
– Pece
Aug 28 at 9:25
No, you're not missing anything.
– Kevin Carlson
Aug 28 at 16:34
add a comment |Â
The closest thing I can think of is that elements of $R$ can be called relative cocell complexes. As you say, most weak factorization systems used in practice are of this form, but the retract closure is so useful-for other purposes than construcying the factorizations-that it seems people usually just go ahead and close $R$ up under retracts.
– Kevin Carlson
Aug 27 at 17:29
@KevinCarlson Thanks, that is more or less what I understood also. However, it seems that the notion defined in my post is a little different from the assertion that $(vphantommathfrak R^perpmathfrak R,(vphantommathfrak R^perpmathfrak R)^perp)$ is a weak factorization system, because in the latter the factorization of some $f$ as $pi$ can yield $p$ as only a retract of an element of $mathfrak R$, while the former require $p$ to be actually in $mathfrak R$. Or am I missing something that makes the two notions equivalent?
– Pece
Aug 28 at 9:25
No, you're not missing anything.
– Kevin Carlson
Aug 28 at 16:34
The closest thing I can think of is that elements of $R$ can be called relative cocell complexes. As you say, most weak factorization systems used in practice are of this form, but the retract closure is so useful-for other purposes than construcying the factorizations-that it seems people usually just go ahead and close $R$ up under retracts.
– Kevin Carlson
Aug 27 at 17:29
The closest thing I can think of is that elements of $R$ can be called relative cocell complexes. As you say, most weak factorization systems used in practice are of this form, but the retract closure is so useful-for other purposes than construcying the factorizations-that it seems people usually just go ahead and close $R$ up under retracts.
– Kevin Carlson
Aug 27 at 17:29
@KevinCarlson Thanks, that is more or less what I understood also. However, it seems that the notion defined in my post is a little different from the assertion that $(vphantommathfrak R^perpmathfrak R,(vphantommathfrak R^perpmathfrak R)^perp)$ is a weak factorization system, because in the latter the factorization of some $f$ as $pi$ can yield $p$ as only a retract of an element of $mathfrak R$, while the former require $p$ to be actually in $mathfrak R$. Or am I missing something that makes the two notions equivalent?
– Pece
Aug 28 at 9:25
@KevinCarlson Thanks, that is more or less what I understood also. However, it seems that the notion defined in my post is a little different from the assertion that $(vphantommathfrak R^perpmathfrak R,(vphantommathfrak R^perpmathfrak R)^perp)$ is a weak factorization system, because in the latter the factorization of some $f$ as $pi$ can yield $p$ as only a retract of an element of $mathfrak R$, while the former require $p$ to be actually in $mathfrak R$. Or am I missing something that makes the two notions equivalent?
– Pece
Aug 28 at 9:25
No, you're not missing anything.
– Kevin Carlson
Aug 28 at 16:34
No, you're not missing anything.
– Kevin Carlson
Aug 28 at 16:34
add a comment |Â
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The closest thing I can think of is that elements of $R$ can be called relative cocell complexes. As you say, most weak factorization systems used in practice are of this form, but the retract closure is so useful-for other purposes than construcying the factorizations-that it seems people usually just go ahead and close $R$ up under retracts.
– Kevin Carlson
Aug 27 at 17:29
@KevinCarlson Thanks, that is more or less what I understood also. However, it seems that the notion defined in my post is a little different from the assertion that $(vphantommathfrak R^perpmathfrak R,(vphantommathfrak R^perpmathfrak R)^perp)$ is a weak factorization system, because in the latter the factorization of some $f$ as $pi$ can yield $p$ as only a retract of an element of $mathfrak R$, while the former require $p$ to be actually in $mathfrak R$. Or am I missing something that makes the two notions equivalent?
– Pece
Aug 28 at 9:25
No, you're not missing anything.
– Kevin Carlson
Aug 28 at 16:34