Limit of weak equivalences in a Bousfield localization












3














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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    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



























      3












      3








      3







      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






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      edited 9 hours ago

























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      Lao-tzu

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      396212






















          2 Answers
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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





























            2














            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

















            • 1




              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
              2 hours ago













            Your Answer





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            2 Answers
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            2 Answers
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            active

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            active

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            active

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            2














            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


























              2














              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
























                2












                2








                2






                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 3 hours ago









                Dmitri Pavlov

                12.9k43482




                12.9k43482























                    2














                    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

















                    • 1




                      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
                      2 hours ago


















                    2














                    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

















                    • 1




                      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
                      2 hours ago
















                    2












                    2








                    2






                    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 2 hours ago









                    Harry Gindi

                    8,799675168




                    8,799675168








                    • 1




                      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
                      2 hours ago
















                    • 1




                      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
                      2 hours ago










                    1




                    1




                    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
                    2 hours ago






                    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
                    2 hours ago




















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