Localization of a model category with respect to a class of maps
$begingroup$
I am little bit lost with the following (standard?) problem in model categories.
Suppose we have a Quillen adjunction between combinatorial model categories:
$$L:Mleftrightarrow N: R $$
and let $(M,Cof,Fib, WE)$ denote the model structure on $M$. I would like to know if it is possible to define a new model structure $(M, Cof, Fib^', WE^')$ such that $f in WE^'$ if and only if $L(f)$ is a weak equivalence in $N$ ?
ct.category-theory homotopy-theory model-categories
edited Nov 11 '18 at 19:42
David White
11.7k460102
asked Nov 11 '18 at 19:12
ABCABC
755
$endgroup$
add a comment |
$begingroup$
I am little bit lost with the following (standard?) problem in model categories.
Suppose we have a Quillen adjunction between combinatorial model categories:
$$L:Mleftrightarrow N: R $$
and let $(M,Cof,Fib, WE)$ denote the model structure on $M$. I would like to know if it is possible to define a new model structure $(M, Cof, Fib^', WE^')$ such that $f in WE^'$ if and only if $L(f)$ is a weak equivalence in $N$ ?
ct.category-theory homotopy-theory model-categories
edited Nov 11 '18 at 19:42
David White
11.7k460102
asked Nov 11 '18 at 19:12
ABCABC
755
$endgroup$
add a comment |
$begingroup$
I am little bit lost with the following (standard?) problem in model categories.
Suppose we have a Quillen adjunction between combinatorial model categories:
$$L:Mleftrightarrow N: R $$
and let $(M,Cof,Fib, WE)$ denote the model structure on $M$. I would like to know if it is possible to define a new model structure $(M, Cof, Fib^', WE^')$ such that $f in WE^'$ if and only if $L(f)$ is a weak equivalence in $N$ ?
ct.category-theory homotopy-theory model-categories
edited Nov 11 '18 at 19:42
David White
11.7k460102
asked Nov 11 '18 at 19:12
ABCABC
755
$endgroup$
I am little bit lost with the following (standard?) problem in model categories.
Suppose we have a Quillen adjunction between combinatorial model categories:
$$L:Mleftrightarrow N: R $$
and let $(M,Cof,Fib, WE)$ denote the model structure on $M$. I would like to know if it is possible to define a new model structure $(M, Cof, Fib^', WE^')$ such that $f in WE^'$ if and only if $L(f)$ is a weak equivalence in $N$ ?
ct.category-theory homotopy-theory model-categories
ct.category-theory homotopy-theory model-categories
edited Nov 11 '18 at 19:42
David White
11.7k460102
asked Nov 11 '18 at 19:12
ABCABC
755
edited Nov 11 '18 at 19:42
David White
11.7k460102
asked Nov 11 '18 at 19:12
ABCABC
755
edited Nov 11 '18 at 19:42
David White
11.7k460102
edited Nov 11 '18 at 19:42
David White
11.7k460102
edited Nov 11 '18 at 19:42
David White
11.7k460102
11.7k460102
asked Nov 11 '18 at 19:12
ABCABC
755
asked Nov 11 '18 at 19:12
ABCABC
755
asked Nov 11 '18 at 19:12
ABCABC
755
755
add a comment |
add a comment |
1 Answer
1
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$begingroup$
This is a very well-studied problem. Hirschhorn's book proves that, if $M$ is left proper and cellular then localization exists with respect to a set of maps. Jeff Smith figured out how to replace cellular by combinatorial; a nice write-up is in Barwick's paper On Left and Right Model Categories and Left and Right Localizations. For a general class of maps, the existence of the localization is conjecturally equivalent to Vopenka's principle. See The orthogonal subcategory problem in homotopy theory by Casacuberta and Chorny (and, also, Definable orthogonality classes in accessible categories are small).
However, if the class of maps is accessible then Lurie shows how to construct the localization, in Section 5.5 of Higher Topos Theory. Since you start with a left Quillen functor between combinatorial model categories, for you this means $L$ should be an accessible functor and the weak equivalences of $N$ should be of small generation (see Prop 5.5.4.16).
answered Nov 11 '18 at 19:41
David WhiteDavid White
11.7k460102
$endgroup$
$begingroup$
I had a look to the Prop 5.5.4.16, it is written in terms of infinite categories. Unfortunately, I'm not used to that language. But If I understand the statement there, then the model categorical interpretation seems to be different in the sense that he uses the derived functor of L and not L... But I'm not sure about my interpretation... :)
$endgroup$
– ABC
Nov 11 '18 at 20:02
$begingroup$
Most of the model categorical proof is formal, and Hirschhorn does a nice job breaking it down into a series of lemmas. Lurie's proof handles the really hard part (as does Smith's proof, and Hirschhorn's via cellularity, based on Bousfield's original argument). So, you can plug Lurie's proof into a model categorical setting just as easily. See Barwick's paper to understand this better.
$endgroup$
– David White
Nov 12 '18 at 14:15
add a comment |
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1 Answer
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$begingroup$
This is a very well-studied problem. Hirschhorn's book proves that, if $M$ is left proper and cellular then localization exists with respect to a set of maps. Jeff Smith figured out how to replace cellular by combinatorial; a nice write-up is in Barwick's paper On Left and Right Model Categories and Left and Right Localizations. For a general class of maps, the existence of the localization is conjecturally equivalent to Vopenka's principle. See The orthogonal subcategory problem in homotopy theory by Casacuberta and Chorny (and, also, Definable orthogonality classes in accessible categories are small).
However, if the class of maps is accessible then Lurie shows how to construct the localization, in Section 5.5 of Higher Topos Theory. Since you start with a left Quillen functor between combinatorial model categories, for you this means $L$ should be an accessible functor and the weak equivalences of $N$ should be of small generation (see Prop 5.5.4.16).
answered Nov 11 '18 at 19:41
David WhiteDavid White
11.7k460102
$endgroup$
$begingroup$
I had a look to the Prop 5.5.4.16, it is written in terms of infinite categories. Unfortunately, I'm not used to that language. But If I understand the statement there, then the model categorical interpretation seems to be different in the sense that he uses the derived functor of L and not L... But I'm not sure about my interpretation... :)
$endgroup$
– ABC
Nov 11 '18 at 20:02
$begingroup$
Most of the model categorical proof is formal, and Hirschhorn does a nice job breaking it down into a series of lemmas. Lurie's proof handles the really hard part (as does Smith's proof, and Hirschhorn's via cellularity, based on Bousfield's original argument). So, you can plug Lurie's proof into a model categorical setting just as easily. See Barwick's paper to understand this better.
$endgroup$
– David White
Nov 12 '18 at 14:15
add a comment |
$begingroup$
This is a very well-studied problem. Hirschhorn's book proves that, if $M$ is left proper and cellular then localization exists with respect to a set of maps. Jeff Smith figured out how to replace cellular by combinatorial; a nice write-up is in Barwick's paper On Left and Right Model Categories and Left and Right Localizations. For a general class of maps, the existence of the localization is conjecturally equivalent to Vopenka's principle. See The orthogonal subcategory problem in homotopy theory by Casacuberta and Chorny (and, also, Definable orthogonality classes in accessible categories are small).
However, if the class of maps is accessible then Lurie shows how to construct the localization, in Section 5.5 of Higher Topos Theory. Since you start with a left Quillen functor between combinatorial model categories, for you this means $L$ should be an accessible functor and the weak equivalences of $N$ should be of small generation (see Prop 5.5.4.16).
answered Nov 11 '18 at 19:41
David WhiteDavid White
11.7k460102
$endgroup$
$begingroup$
I had a look to the Prop 5.5.4.16, it is written in terms of infinite categories. Unfortunately, I'm not used to that language. But If I understand the statement there, then the model categorical interpretation seems to be different in the sense that he uses the derived functor of L and not L... But I'm not sure about my interpretation... :)
$endgroup$
– ABC
Nov 11 '18 at 20:02
$begingroup$
Most of the model categorical proof is formal, and Hirschhorn does a nice job breaking it down into a series of lemmas. Lurie's proof handles the really hard part (as does Smith's proof, and Hirschhorn's via cellularity, based on Bousfield's original argument). So, you can plug Lurie's proof into a model categorical setting just as easily. See Barwick's paper to understand this better.
$endgroup$
– David White
Nov 12 '18 at 14:15
add a comment |
$begingroup$
This is a very well-studied problem. Hirschhorn's book proves that, if $M$ is left proper and cellular then localization exists with respect to a set of maps. Jeff Smith figured out how to replace cellular by combinatorial; a nice write-up is in Barwick's paper On Left and Right Model Categories and Left and Right Localizations. For a general class of maps, the existence of the localization is conjecturally equivalent to Vopenka's principle. See The orthogonal subcategory problem in homotopy theory by Casacuberta and Chorny (and, also, Definable orthogonality classes in accessible categories are small).
However, if the class of maps is accessible then Lurie shows how to construct the localization, in Section 5.5 of Higher Topos Theory. Since you start with a left Quillen functor between combinatorial model categories, for you this means $L$ should be an accessible functor and the weak equivalences of $N$ should be of small generation (see Prop 5.5.4.16).
answered Nov 11 '18 at 19:41
David WhiteDavid White
11.7k460102
$endgroup$
This is a very well-studied problem. Hirschhorn's book proves that, if $M$ is left proper and cellular then localization exists with respect to a set of maps. Jeff Smith figured out how to replace cellular by combinatorial; a nice write-up is in Barwick's paper On Left and Right Model Categories and Left and Right Localizations. For a general class of maps, the existence of the localization is conjecturally equivalent to Vopenka's principle. See The orthogonal subcategory problem in homotopy theory by Casacuberta and Chorny (and, also, Definable orthogonality classes in accessible categories are small).
However, if the class of maps is accessible then Lurie shows how to construct the localization, in Section 5.5 of Higher Topos Theory. Since you start with a left Quillen functor between combinatorial model categories, for you this means $L$ should be an accessible functor and the weak equivalences of $N$ should be of small generation (see Prop 5.5.4.16).
answered Nov 11 '18 at 19:41
David WhiteDavid White
11.7k460102
answered Nov 11 '18 at 19:41
David WhiteDavid White
11.7k460102
answered Nov 11 '18 at 19:41
David WhiteDavid White
11.7k460102
answered Nov 11 '18 at 19:41
David WhiteDavid White
11.7k460102
11.7k460102
$begingroup$
I had a look to the Prop 5.5.4.16, it is written in terms of infinite categories. Unfortunately, I'm not used to that language. But If I understand the statement there, then the model categorical interpretation seems to be different in the sense that he uses the derived functor of L and not L... But I'm not sure about my interpretation... :)
$endgroup$
– ABC
Nov 11 '18 at 20:02
$begingroup$
Most of the model categorical proof is formal, and Hirschhorn does a nice job breaking it down into a series of lemmas. Lurie's proof handles the really hard part (as does Smith's proof, and Hirschhorn's via cellularity, based on Bousfield's original argument). So, you can plug Lurie's proof into a model categorical setting just as easily. See Barwick's paper to understand this better.
$endgroup$
– David White
Nov 12 '18 at 14:15
add a comment |
$begingroup$
I had a look to the Prop 5.5.4.16, it is written in terms of infinite categories. Unfortunately, I'm not used to that language. But If I understand the statement there, then the model categorical interpretation seems to be different in the sense that he uses the derived functor of L and not L... But I'm not sure about my interpretation... :)
$endgroup$
– ABC
Nov 11 '18 at 20:02
$begingroup$
Most of the model categorical proof is formal, and Hirschhorn does a nice job breaking it down into a series of lemmas. Lurie's proof handles the really hard part (as does Smith's proof, and Hirschhorn's via cellularity, based on Bousfield's original argument). So, you can plug Lurie's proof into a model categorical setting just as easily. See Barwick's paper to understand this better.
$endgroup$
– David White
Nov 12 '18 at 14:15
$begingroup$
I had a look to the Prop 5.5.4.16, it is written in terms of infinite categories. Unfortunately, I'm not used to that language. But If I understand the statement there, then the model categorical interpretation seems to be different in the sense that he uses the derived functor of L and not L... But I'm not sure about my interpretation... :)
$endgroup$
– ABC
Nov 11 '18 at 20:02
$begingroup$
I had a look to the Prop 5.5.4.16, it is written in terms of infinite categories. Unfortunately, I'm not used to that language. But If I understand the statement there, then the model categorical interpretation seems to be different in the sense that he uses the derived functor of L and not L... But I'm not sure about my interpretation... :)
$endgroup$
– ABC
Nov 11 '18 at 20:02
$begingroup$
Most of the model categorical proof is formal, and Hirschhorn does a nice job breaking it down into a series of lemmas. Lurie's proof handles the really hard part (as does Smith's proof, and Hirschhorn's via cellularity, based on Bousfield's original argument). So, you can plug Lurie's proof into a model categorical setting just as easily. See Barwick's paper to understand this better.
$endgroup$
– David White
Nov 12 '18 at 14:15
$begingroup$
Most of the model categorical proof is formal, and Hirschhorn does a nice job breaking it down into a series of lemmas. Lurie's proof handles the really hard part (as does Smith's proof, and Hirschhorn's via cellularity, based on Bousfield's original argument). So, you can plug Lurie's proof into a model categorical setting just as easily. See Barwick's paper to understand this better.
$endgroup$
– David White
Nov 12 '18 at 14:15
add a comment |
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