Limit of weak equivalences in a Bousfield localization
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
asked 16 hours ago
Lao-tzu
403312
add a comment |
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
asked 16 hours ago
Lao-tzu
403312
add a comment |
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
asked 16 hours ago
Lao-tzu
403312
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
asked 16 hours ago
Lao-tzu
403312
asked 16 hours ago
Lao-tzu
403312
edited 16 hours ago
asked 16 hours ago
Lao-tzu
403312
asked 16 hours ago
Lao-tzu
403312
asked 16 hours ago
Lao-tzu
403312
403312
add a comment |
add a comment |
2 Answers
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oldest
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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.
answered 10 hours ago
Dmitri Pavlov
13k43482
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
3 hours ago
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
2 hours ago
@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
1 hour ago
@Harry Gindi I find in Morel-Voevodsky, A1-homotopy theory of schemes, at the begining of section 2.1.4, they use a non-categorical definition of homotopy (co)limits for simplicial sheaves—just take section-wise homotopy (co)limits (of simplicial sets) then sheafify, for me this is weird and I don't think it coincides with any of the above things I mentioned in my first comment.
– Lao-tzu
1 hour ago
First question: Yes, the homotopy limit of a diagram in a model category is one way to compute the limit of the diagram in the underlying $infty$-category. Second question: Maybe have a look at Jardine's book on the homotopy theory of sheaves on a site. There is something nontrivial going on.
– Harry Gindi
1 hour ago
|
show 7 more comments
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).
answered 9 hours ago
Harry Gindi
8,829675168
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
9 hours ago
add a comment |
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2 Answers
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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.
answered 10 hours ago
Dmitri Pavlov
13k43482
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
3 hours ago
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
2 hours ago
@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
1 hour ago
@Harry Gindi I find in Morel-Voevodsky, A1-homotopy theory of schemes, at the begining of section 2.1.4, they use a non-categorical definition of homotopy (co)limits for simplicial sheaves—just take section-wise homotopy (co)limits (of simplicial sets) then sheafify, for me this is weird and I don't think it coincides with any of the above things I mentioned in my first comment.
– Lao-tzu
1 hour ago
First question: Yes, the homotopy limit of a diagram in a model category is one way to compute the limit of the diagram in the underlying $infty$-category. Second question: Maybe have a look at Jardine's book on the homotopy theory of sheaves on a site. There is something nontrivial going on.
– Harry Gindi
1 hour ago
|
show 7 more comments
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.
answered 10 hours ago
Dmitri Pavlov
13k43482
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
3 hours ago
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
2 hours ago
@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
1 hour ago
@Harry Gindi I find in Morel-Voevodsky, A1-homotopy theory of schemes, at the begining of section 2.1.4, they use a non-categorical definition of homotopy (co)limits for simplicial sheaves—just take section-wise homotopy (co)limits (of simplicial sets) then sheafify, for me this is weird and I don't think it coincides with any of the above things I mentioned in my first comment.
– Lao-tzu
1 hour ago
First question: Yes, the homotopy limit of a diagram in a model category is one way to compute the limit of the diagram in the underlying $infty$-category. Second question: Maybe have a look at Jardine's book on the homotopy theory of sheaves on a site. There is something nontrivial going on.
– Harry Gindi
1 hour ago
|
show 7 more comments
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.
answered 10 hours ago
Dmitri Pavlov
13k43482
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.
answered 10 hours ago
Dmitri Pavlov
13k43482
answered 10 hours ago
Dmitri Pavlov
13k43482
answered 10 hours ago
Dmitri Pavlov
13k43482
answered 10 hours ago
Dmitri Pavlov
13k43482
13k43482
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
3 hours ago
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
2 hours ago
@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
1 hour ago
@Harry Gindi I find in Morel-Voevodsky, A1-homotopy theory of schemes, at the begining of section 2.1.4, they use a non-categorical definition of homotopy (co)limits for simplicial sheaves—just take section-wise homotopy (co)limits (of simplicial sets) then sheafify, for me this is weird and I don't think it coincides with any of the above things I mentioned in my first comment.
– Lao-tzu
1 hour ago
First question: Yes, the homotopy limit of a diagram in a model category is one way to compute the limit of the diagram in the underlying $infty$-category. Second question: Maybe have a look at Jardine's book on the homotopy theory of sheaves on a site. There is something nontrivial going on.
– Harry Gindi
1 hour ago
|
show 7 more comments
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
3 hours ago
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
2 hours ago
@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
1 hour ago
@Harry Gindi I find in Morel-Voevodsky, A1-homotopy theory of schemes, at the begining of section 2.1.4, they use a non-categorical definition of homotopy (co)limits for simplicial sheaves—just take section-wise homotopy (co)limits (of simplicial sets) then sheafify, for me this is weird and I don't think it coincides with any of the above things I mentioned in my first comment.
– Lao-tzu
1 hour ago
First question: Yes, the homotopy limit of a diagram in a model category is one way to compute the limit of the diagram in the underlying $infty$-category. Second question: Maybe have a look at Jardine's book on the homotopy theory of sheaves on a site. There is something nontrivial going on.
– Harry Gindi
1 hour ago
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
3 hours ago
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
3 hours ago
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
2 hours ago
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
2 hours ago
@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
1 hour ago
@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
1 hour ago
@Harry Gindi I find in Morel-Voevodsky, A1-homotopy theory of schemes, at the begining of section 2.1.4, they use a non-categorical definition of homotopy (co)limits for simplicial sheaves—just take section-wise homotopy (co)limits (of simplicial sets) then sheafify, for me this is weird and I don't think it coincides with any of the above things I mentioned in my first comment.
– Lao-tzu
1 hour ago
@Harry Gindi I find in Morel-Voevodsky, A1-homotopy theory of schemes, at the begining of section 2.1.4, they use a non-categorical definition of homotopy (co)limits for simplicial sheaves—just take section-wise homotopy (co)limits (of simplicial sets) then sheafify, for me this is weird and I don't think it coincides with any of the above things I mentioned in my first comment.
– Lao-tzu
1 hour ago
First question: Yes, the homotopy limit of a diagram in a model category is one way to compute the limit of the diagram in the underlying $infty$-category. Second question: Maybe have a look at Jardine's book on the homotopy theory of sheaves on a site. There is something nontrivial going on.
– Harry Gindi
1 hour ago
First question: Yes, the homotopy limit of a diagram in a model category is one way to compute the limit of the diagram in the underlying $infty$-category. Second question: Maybe have a look at Jardine's book on the homotopy theory of sheaves on a site. There is something nontrivial going on.
– Harry Gindi
1 hour ago
|
show 7 more comments
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).
answered 9 hours ago
Harry Gindi
8,829675168
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
9 hours ago
add a comment |
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).
answered 9 hours ago
Harry Gindi
8,829675168
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
9 hours ago
add a comment |
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).
answered 9 hours ago
Harry Gindi
8,829675168
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).
answered 9 hours ago
Harry Gindi
8,829675168
answered 9 hours ago
Harry Gindi
8,829675168
answered 9 hours ago
Harry Gindi
8,829675168
answered 9 hours ago
Harry Gindi
8,829675168
8,829675168
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
9 hours ago
add a comment |
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
9 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
9 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
9 hours ago
add a comment |
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