Why Bousfield localization preserves homotopy pull-backs?

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Studying chromatic homotopy theory I encountered the chromatic fracture square
$requireAMScd$
beginCD
L_K(n) vee K(m)X @>>> L_K(m)X\
@V V V @VV V\
L_K(n)X @>>> L_K(n)L_K(m)X
endCD
for $n<m$. Then I was told that applying a Bousfield localization functor we preserve the homotopy pull-back, this is useful since applying $L_K(t)$ for $n <t<m$ we get the equality $L_K(t)L_K(n)vee K(m)X= L_K(t)L_K(m)X $
since the lower terms become zero.

I would like to know a proof of this claim. Also a reference for this fact is well accepted.
Thank you for any help.

stable-homotopy-theory pullback

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edited Jul 20 at 8:07

asked Jul 18 at 17:13

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up vote
0
down vote

favorite

Studying chromatic homotopy theory I encountered the chromatic fracture square
$requireAMScd$
beginCD
L_K(n) vee K(m)X @>>> L_K(m)X\
@V V V @VV V\
L_K(n)X @>>> L_K(n)L_K(m)X
endCD
for $n<m$. Then I was told that applying a Bousfield localization functor we preserve the homotopy pull-back, this is useful since applying $L_K(t)$ for $n <t<m$ we get the equality $L_K(t)L_K(n)vee K(m)X= L_K(t)L_K(m)X $
since the lower terms become zero.

I would like to know a proof of this claim. Also a reference for this fact is well accepted.
Thank you for any help.

stable-homotopy-theory pullback

share|cite|improve this question

edited Jul 20 at 8:07

asked Jul 18 at 17:13

N.B.

433212

add a comment | 

up vote
0
down vote

favorite

up vote
0
down vote

favorite

Studying chromatic homotopy theory I encountered the chromatic fracture square
$requireAMScd$
beginCD
L_K(n) vee K(m)X @>>> L_K(m)X\
@V V V @VV V\
L_K(n)X @>>> L_K(n)L_K(m)X
endCD
for $n<m$. Then I was told that applying a Bousfield localization functor we preserve the homotopy pull-back, this is useful since applying $L_K(t)$ for $n <t<m$ we get the equality $L_K(t)L_K(n)vee K(m)X= L_K(t)L_K(m)X $
since the lower terms become zero.

I would like to know a proof of this claim. Also a reference for this fact is well accepted.
Thank you for any help.

stable-homotopy-theory pullback

share|cite|improve this question

edited Jul 20 at 8:07

asked Jul 18 at 17:13

N.B.

433212

Studying chromatic homotopy theory I encountered the chromatic fracture square
$requireAMScd$
beginCD
L_K(n) vee K(m)X @>>> L_K(m)X\
@V V V @VV V\
L_K(n)X @>>> L_K(n)L_K(m)X
endCD
for $n<m$. Then I was told that applying a Bousfield localization functor we preserve the homotopy pull-back, this is useful since applying $L_K(t)$ for $n <t<m$ we get the equality $L_K(t)L_K(n)vee K(m)X= L_K(t)L_K(m)X $
since the lower terms become zero.

I would like to know a proof of this claim. Also a reference for this fact is well accepted.
Thank you for any help.

stable-homotopy-theory pullback

share|cite|improve this question

edited Jul 20 at 8:07

asked Jul 18 at 17:13

N.B.

433212

share|cite|improve this question

share|cite|improve this question

edited Jul 20 at 8:07

edited Jul 20 at 8:07

edited Jul 20 at 8:07

asked Jul 18 at 17:13

N.B.

433212

asked Jul 18 at 17:13

N.B.

433212

asked Jul 18 at 17:13

N.B.

433212

433212

add a comment | 

add a comment | 

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I just need to observe that the Bousfield localization, being a left adjoint, preserves homotopy colimits. Now by stability of the stable homotopy category homotopy pull-backs and pushouts coincide.

share|cite|improve this answer

answered Jul 20 at 12:46

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1 Answer
1

active

oldest

votes

1 Answer
1

active

oldest

votes

active

oldest

votes

active

oldest

votes

up vote
0
down vote

I just need to observe that the Bousfield localization, being a left adjoint, preserves homotopy colimits. Now by stability of the stable homotopy category homotopy pull-backs and pushouts coincide.

share|cite|improve this answer

answered Jul 20 at 12:46

N.B.

433212

add a comment | 

up vote
0
down vote

I just need to observe that the Bousfield localization, being a left adjoint, preserves homotopy colimits. Now by stability of the stable homotopy category homotopy pull-backs and pushouts coincide.

share|cite|improve this answer

answered Jul 20 at 12:46

N.B.

433212

add a comment | 

up vote
0
down vote

up vote
0
down vote

I just need to observe that the Bousfield localization, being a left adjoint, preserves homotopy colimits. Now by stability of the stable homotopy category homotopy pull-backs and pushouts coincide.

share|cite|improve this answer

answered Jul 20 at 12:46

N.B.

433212

I just need to observe that the Bousfield localization, being a left adjoint, preserves homotopy colimits. Now by stability of the stable homotopy category homotopy pull-backs and pushouts coincide.

share|cite|improve this answer

answered Jul 20 at 12:46

N.B.

433212

share|cite|improve this answer

share|cite|improve this answer

answered Jul 20 at 12:46

N.B.

433212

answered Jul 20 at 12:46

N.B.

433212

answered Jul 20 at 12:46

N.B.

433212

433212

add a comment | 

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