Limit of weak equivalences in a Bousfield localization












5














Let $M$ be a model category and $C$ a class of maps in it, and assume the left Bousfield localization $L_CM$ exists. Suppose we are given sequences of maps $(p_{n+1}: X_{n+1}to X_n), (q_{n+1}: Y_{n+1}to Y_n), (f_n: X_nto Y_n), n=0, 1,ldots$ with $q_{n+1}f_{n+1}=f_np_{n+1}$, so we get a ladder of commutative squares. If each $p_n$ is a fibration of fibrants in $M$, each $q_n$ is a fibration of fibrants in $L_CM$, and each $f_n$ is a weak equivalence in $L_CM$, can we conclude that the limit map $lim f_n$ is also a weak equivalence in $L_CM$?



For the notion of left Bousfield localization, see Hirschhorn, Model categories and their localizations, chapter 3, 4. See Proposition 15.10.12 in that book for a similar result, my question is by weakening the assumption as well as the conclusion. You may add suitable and reasonable conditions—like simplicial, properness, cofibrantly generated, etc.—if needed.






homotopy-theory model-categories bousfield-localization







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    5














    Let $M$ be a model category and $C$ a class of maps in it, and assume the left Bousfield localization $L_CM$ exists. Suppose we are given sequences of maps $(p_{n+1}: X_{n+1}to X_n), (q_{n+1}: Y_{n+1}to Y_n), (f_n: X_nto Y_n), n=0, 1,ldots$ with $q_{n+1}f_{n+1}=f_np_{n+1}$, so we get a ladder of commutative squares. If each $p_n$ is a fibration of fibrants in $M$, each $q_n$ is a fibration of fibrants in $L_CM$, and each $f_n$ is a weak equivalence in $L_CM$, can we conclude that the limit map $lim f_n$ is also a weak equivalence in $L_CM$?



    For the notion of left Bousfield localization, see Hirschhorn, Model categories and their localizations, chapter 3, 4. See Proposition 15.10.12 in that book for a similar result, my question is by weakening the assumption as well as the conclusion. You may add suitable and reasonable conditions—like simplicial, properness, cofibrantly generated, etc.—if needed.






    homotopy-theory model-categories bousfield-localization







    share|cite|improve this question



























      5












      5








      5


      1





      Let $M$ be a model category and $C$ a class of maps in it, and assume the left Bousfield localization $L_CM$ exists. Suppose we are given sequences of maps $(p_{n+1}: X_{n+1}to X_n), (q_{n+1}: Y_{n+1}to Y_n), (f_n: X_nto Y_n), n=0, 1,ldots$ with $q_{n+1}f_{n+1}=f_np_{n+1}$, so we get a ladder of commutative squares. If each $p_n$ is a fibration of fibrants in $M$, each $q_n$ is a fibration of fibrants in $L_CM$, and each $f_n$ is a weak equivalence in $L_CM$, can we conclude that the limit map $lim f_n$ is also a weak equivalence in $L_CM$?



      For the notion of left Bousfield localization, see Hirschhorn, Model categories and their localizations, chapter 3, 4. See Proposition 15.10.12 in that book for a similar result, my question is by weakening the assumption as well as the conclusion. You may add suitable and reasonable conditions—like simplicial, properness, cofibrantly generated, etc.—if needed.






      homotopy-theory model-categories bousfield-localization







      share|cite|improve this question















      Let $M$ be a model category and $C$ a class of maps in it, and assume the left Bousfield localization $L_CM$ exists. Suppose we are given sequences of maps $(p_{n+1}: X_{n+1}to X_n), (q_{n+1}: Y_{n+1}to Y_n), (f_n: X_nto Y_n), n=0, 1,ldots$ with $q_{n+1}f_{n+1}=f_np_{n+1}$, so we get a ladder of commutative squares. If each $p_n$ is a fibration of fibrants in $M$, each $q_n$ is a fibration of fibrants in $L_CM$, and each $f_n$ is a weak equivalence in $L_CM$, can we conclude that the limit map $lim f_n$ is also a weak equivalence in $L_CM$?



      For the notion of left Bousfield localization, see Hirschhorn, Model categories and their localizations, chapter 3, 4. See Proposition 15.10.12 in that book for a similar result, my question is by weakening the assumption as well as the conclusion. You may add suitable and reasonable conditions—like simplicial, properness, cofibrantly generated, etc.—if needed.






      homotopy-theory model-categories bousfield-localization




      homotopy-theory model-categories bousfield-localization






      share|cite|improve this question















      share|cite|improve this question













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      edited Dec 25 at 22:25

























      asked Dec 25 at 22:19









      Lao-tzu

      408312




      408312






















          2 Answers
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          active

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          3














          No. For a counterexample to your claim, consider the model category M
          of simplicial presheaves on a small site S equipped with the projective
          model structure.
          Its fibrant objects are presheaves of Kan complexes.
          If C is the set of Čech covers of S, then L_C(M) is the local projective
          model structure on simplicial presheaves.
          Its fibrant objects are presheaves of Kan complexes that satisfy homotopy descent.
          A weak equivalence from a fibrant object in M to a fibrant object in L_C(M)
          is a homotopy sheafification map.
          Furthermore, the limit of p and q is a homotopy limit in M,
          so lim f_n is a weak equivalence if and only if the homotopy sheafification
          functor preserves homotopy limits of towers.
          This is false for arbitrary sites.






          share|cite|improve this answer

















          • 1




            What's your def of homotopy (co)limits? I know it's discussed in the last two chapters of Hirschhorn's book by formulas. But his homotopy limits has not fully homotopy invariance—the objects involved in the diagrams should be fibrant in the model structure. But someone else will take functorial fibrant replacement before using the formulas, which I know is weakly equivalent to the right derived functor of $lim$ in nice cases (and someone always use this right derived functor as def of homotopy limits). Do you know what is the most "correct" and/or most accepted def?
            – Lao-tzu
            Dec 26 at 11:38












          • The difficulty of defining homotopy limits and colimits by universal property is one of the main motivations behind $infty$-categories. Without such a definition, all 'definitions' are actually nontrivially equivalent computations.
            – Harry Gindi
            Dec 26 at 12:32










          • @Harry Gindi Is the notion of homotopy limits in a model category just a special case of limits for $infty$-categories (when taking the $infty$-category associated to the given model category) or there is some relation you can explain?
            – Lao-tzu
            Dec 26 at 13:11






          • 1




            @DimitriPavlov I found unwinding the functorial case in the relative category model to be highly involved. It is a large portion of the book of Dwyer-Hirschhorn-Kan-Smith. Usually you can punt the question and pop through to the hammock localization and then define it in terms of hom-wise simplicial holims, but in the Quasicategory case, the non-functorial definition amounts to nothing more than a terminal object of a slice over the diagram.
            – Harry Gindi
            Dec 26 at 15:31






          • 1




            @HarryGindi: Sure, the unwinding can be involved, but notice that your original claim was not about the complexity of computations, but rather about the complexity of definitions, a different notion.
            – Dmitri Pavlov
            Dec 26 at 15:38



















          7














          In the language of $infty$-categories, which makes it a bit clearer, this is asking for the reflector (left adjoint) of the inclusion of a reflective subcategory to preserve filtered limits. This isn't true for ordinary categories, and there is also no reason to expect it to be true for $infty$-categories.



          Hirschhorn's Proposition 15.10.12 says that the homotopy limit of a tower of fibrations can be computed as the ordinary limit. Your modification asks for this homotopy limit to be preserved by the reflector (localization functor).






          share|cite|improve this answer

















          • 2




            As for a condition for this to hold, it's true if the $f_n$ are $M$-equivalences, but this makes the statement trivial, since it implies that the $X_n$ are $C$-local and forces the $p_n$ to be local fibrations (by the usual results about left Bousfield localization, being $C$-local is invariant under $M$-equivalence and $M$-fibrations between local objects are also local fibrations.)
            – Harry Gindi
            Dec 26 at 5:35













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          No. For a counterexample to your claim, consider the model category M
          of simplicial presheaves on a small site S equipped with the projective
          model structure.
          Its fibrant objects are presheaves of Kan complexes.
          If C is the set of Čech covers of S, then L_C(M) is the local projective
          model structure on simplicial presheaves.
          Its fibrant objects are presheaves of Kan complexes that satisfy homotopy descent.
          A weak equivalence from a fibrant object in M to a fibrant object in L_C(M)
          is a homotopy sheafification map.
          Furthermore, the limit of p and q is a homotopy limit in M,
          so lim f_n is a weak equivalence if and only if the homotopy sheafification
          functor preserves homotopy limits of towers.
          This is false for arbitrary sites.






          share|cite|improve this answer

















          • 1




            What's your def of homotopy (co)limits? I know it's discussed in the last two chapters of Hirschhorn's book by formulas. But his homotopy limits has not fully homotopy invariance—the objects involved in the diagrams should be fibrant in the model structure. But someone else will take functorial fibrant replacement before using the formulas, which I know is weakly equivalent to the right derived functor of $lim$ in nice cases (and someone always use this right derived functor as def of homotopy limits). Do you know what is the most "correct" and/or most accepted def?
            – Lao-tzu
            Dec 26 at 11:38












          • The difficulty of defining homotopy limits and colimits by universal property is one of the main motivations behind $infty$-categories. Without such a definition, all 'definitions' are actually nontrivially equivalent computations.
            – Harry Gindi
            Dec 26 at 12:32










          • @Harry Gindi Is the notion of homotopy limits in a model category just a special case of limits for $infty$-categories (when taking the $infty$-category associated to the given model category) or there is some relation you can explain?
            – Lao-tzu
            Dec 26 at 13:11






          • 1




            @DimitriPavlov I found unwinding the functorial case in the relative category model to be highly involved. It is a large portion of the book of Dwyer-Hirschhorn-Kan-Smith. Usually you can punt the question and pop through to the hammock localization and then define it in terms of hom-wise simplicial holims, but in the Quasicategory case, the non-functorial definition amounts to nothing more than a terminal object of a slice over the diagram.
            – Harry Gindi
            Dec 26 at 15:31






          • 1




            @HarryGindi: Sure, the unwinding can be involved, but notice that your original claim was not about the complexity of computations, but rather about the complexity of definitions, a different notion.
            – Dmitri Pavlov
            Dec 26 at 15:38
















          3














          No. For a counterexample to your claim, consider the model category M
          of simplicial presheaves on a small site S equipped with the projective
          model structure.
          Its fibrant objects are presheaves of Kan complexes.
          If C is the set of Čech covers of S, then L_C(M) is the local projective
          model structure on simplicial presheaves.
          Its fibrant objects are presheaves of Kan complexes that satisfy homotopy descent.
          A weak equivalence from a fibrant object in M to a fibrant object in L_C(M)
          is a homotopy sheafification map.
          Furthermore, the limit of p and q is a homotopy limit in M,
          so lim f_n is a weak equivalence if and only if the homotopy sheafification
          functor preserves homotopy limits of towers.
          This is false for arbitrary sites.






          share|cite|improve this answer

















          • 1




            What's your def of homotopy (co)limits? I know it's discussed in the last two chapters of Hirschhorn's book by formulas. But his homotopy limits has not fully homotopy invariance—the objects involved in the diagrams should be fibrant in the model structure. But someone else will take functorial fibrant replacement before using the formulas, which I know is weakly equivalent to the right derived functor of $lim$ in nice cases (and someone always use this right derived functor as def of homotopy limits). Do you know what is the most "correct" and/or most accepted def?
            – Lao-tzu
            Dec 26 at 11:38












          • The difficulty of defining homotopy limits and colimits by universal property is one of the main motivations behind $infty$-categories. Without such a definition, all 'definitions' are actually nontrivially equivalent computations.
            – Harry Gindi
            Dec 26 at 12:32










          • @Harry Gindi Is the notion of homotopy limits in a model category just a special case of limits for $infty$-categories (when taking the $infty$-category associated to the given model category) or there is some relation you can explain?
            – Lao-tzu
            Dec 26 at 13:11






          • 1




            @DimitriPavlov I found unwinding the functorial case in the relative category model to be highly involved. It is a large portion of the book of Dwyer-Hirschhorn-Kan-Smith. Usually you can punt the question and pop through to the hammock localization and then define it in terms of hom-wise simplicial holims, but in the Quasicategory case, the non-functorial definition amounts to nothing more than a terminal object of a slice over the diagram.
            – Harry Gindi
            Dec 26 at 15:31






          • 1




            @HarryGindi: Sure, the unwinding can be involved, but notice that your original claim was not about the complexity of computations, but rather about the complexity of definitions, a different notion.
            – Dmitri Pavlov
            Dec 26 at 15:38














          3












          3








          3






          No. For a counterexample to your claim, consider the model category M
          of simplicial presheaves on a small site S equipped with the projective
          model structure.
          Its fibrant objects are presheaves of Kan complexes.
          If C is the set of Čech covers of S, then L_C(M) is the local projective
          model structure on simplicial presheaves.
          Its fibrant objects are presheaves of Kan complexes that satisfy homotopy descent.
          A weak equivalence from a fibrant object in M to a fibrant object in L_C(M)
          is a homotopy sheafification map.
          Furthermore, the limit of p and q is a homotopy limit in M,
          so lim f_n is a weak equivalence if and only if the homotopy sheafification
          functor preserves homotopy limits of towers.
          This is false for arbitrary sites.






          share|cite|improve this answer












          No. For a counterexample to your claim, consider the model category M
          of simplicial presheaves on a small site S equipped with the projective
          model structure.
          Its fibrant objects are presheaves of Kan complexes.
          If C is the set of Čech covers of S, then L_C(M) is the local projective
          model structure on simplicial presheaves.
          Its fibrant objects are presheaves of Kan complexes that satisfy homotopy descent.
          A weak equivalence from a fibrant object in M to a fibrant object in L_C(M)
          is a homotopy sheafification map.
          Furthermore, the limit of p and q is a homotopy limit in M,
          so lim f_n is a weak equivalence if and only if the homotopy sheafification
          functor preserves homotopy limits of towers.
          This is false for arbitrary sites.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 26 at 4:37









          Dmitri Pavlov

          13k43482




          13k43482








          • 1




            What's your def of homotopy (co)limits? I know it's discussed in the last two chapters of Hirschhorn's book by formulas. But his homotopy limits has not fully homotopy invariance—the objects involved in the diagrams should be fibrant in the model structure. But someone else will take functorial fibrant replacement before using the formulas, which I know is weakly equivalent to the right derived functor of $lim$ in nice cases (and someone always use this right derived functor as def of homotopy limits). Do you know what is the most "correct" and/or most accepted def?
            – Lao-tzu
            Dec 26 at 11:38












          • The difficulty of defining homotopy limits and colimits by universal property is one of the main motivations behind $infty$-categories. Without such a definition, all 'definitions' are actually nontrivially equivalent computations.
            – Harry Gindi
            Dec 26 at 12:32










          • @Harry Gindi Is the notion of homotopy limits in a model category just a special case of limits for $infty$-categories (when taking the $infty$-category associated to the given model category) or there is some relation you can explain?
            – Lao-tzu
            Dec 26 at 13:11






          • 1




            @DimitriPavlov I found unwinding the functorial case in the relative category model to be highly involved. It is a large portion of the book of Dwyer-Hirschhorn-Kan-Smith. Usually you can punt the question and pop through to the hammock localization and then define it in terms of hom-wise simplicial holims, but in the Quasicategory case, the non-functorial definition amounts to nothing more than a terminal object of a slice over the diagram.
            – Harry Gindi
            Dec 26 at 15:31






          • 1




            @HarryGindi: Sure, the unwinding can be involved, but notice that your original claim was not about the complexity of computations, but rather about the complexity of definitions, a different notion.
            – Dmitri Pavlov
            Dec 26 at 15:38














          • 1




            What's your def of homotopy (co)limits? I know it's discussed in the last two chapters of Hirschhorn's book by formulas. But his homotopy limits has not fully homotopy invariance—the objects involved in the diagrams should be fibrant in the model structure. But someone else will take functorial fibrant replacement before using the formulas, which I know is weakly equivalent to the right derived functor of $lim$ in nice cases (and someone always use this right derived functor as def of homotopy limits). Do you know what is the most "correct" and/or most accepted def?
            – Lao-tzu
            Dec 26 at 11:38












          • The difficulty of defining homotopy limits and colimits by universal property is one of the main motivations behind $infty$-categories. Without such a definition, all 'definitions' are actually nontrivially equivalent computations.
            – Harry Gindi
            Dec 26 at 12:32










          • @Harry Gindi Is the notion of homotopy limits in a model category just a special case of limits for $infty$-categories (when taking the $infty$-category associated to the given model category) or there is some relation you can explain?
            – Lao-tzu
            Dec 26 at 13:11






          • 1




            @DimitriPavlov I found unwinding the functorial case in the relative category model to be highly involved. It is a large portion of the book of Dwyer-Hirschhorn-Kan-Smith. Usually you can punt the question and pop through to the hammock localization and then define it in terms of hom-wise simplicial holims, but in the Quasicategory case, the non-functorial definition amounts to nothing more than a terminal object of a slice over the diagram.
            – Harry Gindi
            Dec 26 at 15:31






          • 1




            @HarryGindi: Sure, the unwinding can be involved, but notice that your original claim was not about the complexity of computations, but rather about the complexity of definitions, a different notion.
            – Dmitri Pavlov
            Dec 26 at 15:38








          1




          1




          What's your def of homotopy (co)limits? I know it's discussed in the last two chapters of Hirschhorn's book by formulas. But his homotopy limits has not fully homotopy invariance—the objects involved in the diagrams should be fibrant in the model structure. But someone else will take functorial fibrant replacement before using the formulas, which I know is weakly equivalent to the right derived functor of $lim$ in nice cases (and someone always use this right derived functor as def of homotopy limits). Do you know what is the most "correct" and/or most accepted def?
          – Lao-tzu
          Dec 26 at 11:38






          What's your def of homotopy (co)limits? I know it's discussed in the last two chapters of Hirschhorn's book by formulas. But his homotopy limits has not fully homotopy invariance—the objects involved in the diagrams should be fibrant in the model structure. But someone else will take functorial fibrant replacement before using the formulas, which I know is weakly equivalent to the right derived functor of $lim$ in nice cases (and someone always use this right derived functor as def of homotopy limits). Do you know what is the most "correct" and/or most accepted def?
          – Lao-tzu
          Dec 26 at 11:38














          The difficulty of defining homotopy limits and colimits by universal property is one of the main motivations behind $infty$-categories. Without such a definition, all 'definitions' are actually nontrivially equivalent computations.
          – Harry Gindi
          Dec 26 at 12:32




          The difficulty of defining homotopy limits and colimits by universal property is one of the main motivations behind $infty$-categories. Without such a definition, all 'definitions' are actually nontrivially equivalent computations.
          – Harry Gindi
          Dec 26 at 12:32












          @Harry Gindi Is the notion of homotopy limits in a model category just a special case of limits for $infty$-categories (when taking the $infty$-category associated to the given model category) or there is some relation you can explain?
          – Lao-tzu
          Dec 26 at 13:11




          @Harry Gindi Is the notion of homotopy limits in a model category just a special case of limits for $infty$-categories (when taking the $infty$-category associated to the given model category) or there is some relation you can explain?
          – Lao-tzu
          Dec 26 at 13:11




          1




          1




          @DimitriPavlov I found unwinding the functorial case in the relative category model to be highly involved. It is a large portion of the book of Dwyer-Hirschhorn-Kan-Smith. Usually you can punt the question and pop through to the hammock localization and then define it in terms of hom-wise simplicial holims, but in the Quasicategory case, the non-functorial definition amounts to nothing more than a terminal object of a slice over the diagram.
          – Harry Gindi
          Dec 26 at 15:31




          @DimitriPavlov I found unwinding the functorial case in the relative category model to be highly involved. It is a large portion of the book of Dwyer-Hirschhorn-Kan-Smith. Usually you can punt the question and pop through to the hammock localization and then define it in terms of hom-wise simplicial holims, but in the Quasicategory case, the non-functorial definition amounts to nothing more than a terminal object of a slice over the diagram.
          – Harry Gindi
          Dec 26 at 15:31




          1




          1




          @HarryGindi: Sure, the unwinding can be involved, but notice that your original claim was not about the complexity of computations, but rather about the complexity of definitions, a different notion.
          – Dmitri Pavlov
          Dec 26 at 15:38




          @HarryGindi: Sure, the unwinding can be involved, but notice that your original claim was not about the complexity of computations, but rather about the complexity of definitions, a different notion.
          – Dmitri Pavlov
          Dec 26 at 15:38











          7














          In the language of $infty$-categories, which makes it a bit clearer, this is asking for the reflector (left adjoint) of the inclusion of a reflective subcategory to preserve filtered limits. This isn't true for ordinary categories, and there is also no reason to expect it to be true for $infty$-categories.



          Hirschhorn's Proposition 15.10.12 says that the homotopy limit of a tower of fibrations can be computed as the ordinary limit. Your modification asks for this homotopy limit to be preserved by the reflector (localization functor).






          share|cite|improve this answer

















          • 2




            As for a condition for this to hold, it's true if the $f_n$ are $M$-equivalences, but this makes the statement trivial, since it implies that the $X_n$ are $C$-local and forces the $p_n$ to be local fibrations (by the usual results about left Bousfield localization, being $C$-local is invariant under $M$-equivalence and $M$-fibrations between local objects are also local fibrations.)
            – Harry Gindi
            Dec 26 at 5:35


















          7














          In the language of $infty$-categories, which makes it a bit clearer, this is asking for the reflector (left adjoint) of the inclusion of a reflective subcategory to preserve filtered limits. This isn't true for ordinary categories, and there is also no reason to expect it to be true for $infty$-categories.



          Hirschhorn's Proposition 15.10.12 says that the homotopy limit of a tower of fibrations can be computed as the ordinary limit. Your modification asks for this homotopy limit to be preserved by the reflector (localization functor).






          share|cite|improve this answer

















          • 2




            As for a condition for this to hold, it's true if the $f_n$ are $M$-equivalences, but this makes the statement trivial, since it implies that the $X_n$ are $C$-local and forces the $p_n$ to be local fibrations (by the usual results about left Bousfield localization, being $C$-local is invariant under $M$-equivalence and $M$-fibrations between local objects are also local fibrations.)
            – Harry Gindi
            Dec 26 at 5:35
















          7












          7








          7






          In the language of $infty$-categories, which makes it a bit clearer, this is asking for the reflector (left adjoint) of the inclusion of a reflective subcategory to preserve filtered limits. This isn't true for ordinary categories, and there is also no reason to expect it to be true for $infty$-categories.



          Hirschhorn's Proposition 15.10.12 says that the homotopy limit of a tower of fibrations can be computed as the ordinary limit. Your modification asks for this homotopy limit to be preserved by the reflector (localization functor).






          share|cite|improve this answer












          In the language of $infty$-categories, which makes it a bit clearer, this is asking for the reflector (left adjoint) of the inclusion of a reflective subcategory to preserve filtered limits. This isn't true for ordinary categories, and there is also no reason to expect it to be true for $infty$-categories.



          Hirschhorn's Proposition 15.10.12 says that the homotopy limit of a tower of fibrations can be computed as the ordinary limit. Your modification asks for this homotopy limit to be preserved by the reflector (localization functor).







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 26 at 5:18









          Harry Gindi

          8,849676168




          8,849676168








          • 2




            As for a condition for this to hold, it's true if the $f_n$ are $M$-equivalences, but this makes the statement trivial, since it implies that the $X_n$ are $C$-local and forces the $p_n$ to be local fibrations (by the usual results about left Bousfield localization, being $C$-local is invariant under $M$-equivalence and $M$-fibrations between local objects are also local fibrations.)
            – Harry Gindi
            Dec 26 at 5:35
















          • 2




            As for a condition for this to hold, it's true if the $f_n$ are $M$-equivalences, but this makes the statement trivial, since it implies that the $X_n$ are $C$-local and forces the $p_n$ to be local fibrations (by the usual results about left Bousfield localization, being $C$-local is invariant under $M$-equivalence and $M$-fibrations between local objects are also local fibrations.)
            – Harry Gindi
            Dec 26 at 5:35










          2




          2




          As for a condition for this to hold, it's true if the $f_n$ are $M$-equivalences, but this makes the statement trivial, since it implies that the $X_n$ are $C$-local and forces the $p_n$ to be local fibrations (by the usual results about left Bousfield localization, being $C$-local is invariant under $M$-equivalence and $M$-fibrations between local objects are also local fibrations.)
          – Harry Gindi
          Dec 26 at 5:35






          As for a condition for this to hold, it's true if the $f_n$ are $M$-equivalences, but this makes the statement trivial, since it implies that the $X_n$ are $C$-local and forces the $p_n$ to be local fibrations (by the usual results about left Bousfield localization, being $C$-local is invariant under $M$-equivalence and $M$-fibrations between local objects are also local fibrations.)
          – Harry Gindi
          Dec 26 at 5:35




















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