(Co-)fibrations in Top and CGWH
$begingroup$
Suppose that you have a map $i: Arightarrow X$ between CGWH (compactly generated weakly Hausdorff) spaces. It is true that if it is a cofibration in CGWH (the category of CGWH spaces), then it is a cofibration in Top (the category of topological spaces)?
What about Hurewicz fibrations?
general-topology algebraic-topology homotopy-theory
asked Dec 6 '18 at 15:14
GreggGregg
9318
$endgroup$
add a comment |
$begingroup$
Suppose that you have a map $i: Arightarrow X$ between CGWH (compactly generated weakly Hausdorff) spaces. It is true that if it is a cofibration in CGWH (the category of CGWH spaces), then it is a cofibration in Top (the category of topological spaces)?
What about Hurewicz fibrations?
general-topology algebraic-topology homotopy-theory
asked Dec 6 '18 at 15:14
GreggGregg
9318
$endgroup$
$begingroup$
Interesting question. You certainly know Neil Stricklands's paper "The category of CGWH spaces" neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf. It is easy to see that the product of weakly Hausdorff spaces it weakly Hausdorff. Hence by Strickland 2.6, if $X$ is CGWH, then so is $X times I$. Using Strickland 1.10 we see that if a map $i : A to X$ in CGWH is a cofibration in CG, then it is also one in Top. However, there is a still a gap: If $i$ is a cofibration in CGWH, is it one in CG?
$endgroup$
– Paul Frost
Dec 6 '18 at 16:20
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@PaulFrost Actually, I didn’t know about this paper before. I will have a look, thank you
$endgroup$
– Gregg
Dec 6 '18 at 18:02
$begingroup$
Concerning the last question in my post, the answer is negative. It is known that any Serre fibration between CW complexes is a Hurewicz fibration in CGWH, but it may be not Hurewicz in Top (for a proof and relevant counterexample, see M. Steinberger and J. West. Covering homotopy properties of maps between CW complexes or ANRs. Proc. Amer. Math. Soc. 92(1984), 573-577)
$endgroup$
– Gregg
Dec 6 '18 at 18:06
1
$begingroup$
If $i(A)$ is closed in $X$, then $i$ is a cofibration in Top iff $Z = X times 0 cup i(A) times I$ is a retract of $X times I$. It suffices to know that $i$ is a cofibration in CGWH to show that $Z$ is a retract of $X times I$: $Z$ is closed in $X times I$, hence a CGWH space.
$endgroup$
– Paul Frost
Dec 6 '18 at 19:54
$begingroup$
@PaulFrost Actually, $i$ is a cofibration in Top iff the canonical map $l: M_ito Xtimes I$ has a retract iff $Z$ is a retract of $Xtimes I$, so the closedness assumption is not necessary. (However, it can be shown that a cofibration in CGWH is always a closed embedding.) But knowing that a map is cofibration in CGWH is indeed enough to conlude that $l$ has a retract (because the pushout of a diagram in which at least one of the maps is a closed embedding is in CGWH). Thank you!
$endgroup$
– Gregg
Dec 8 '18 at 19:40
add a comment |
$begingroup$
Suppose that you have a map $i: Arightarrow X$ between CGWH (compactly generated weakly Hausdorff) spaces. It is true that if it is a cofibration in CGWH (the category of CGWH spaces), then it is a cofibration in Top (the category of topological spaces)?
What about Hurewicz fibrations?
general-topology algebraic-topology homotopy-theory
asked Dec 6 '18 at 15:14
GreggGregg
9318
$endgroup$
Suppose that you have a map $i: Arightarrow X$ between CGWH (compactly generated weakly Hausdorff) spaces. It is true that if it is a cofibration in CGWH (the category of CGWH spaces), then it is a cofibration in Top (the category of topological spaces)?
What about Hurewicz fibrations?
general-topology algebraic-topology homotopy-theory
general-topology algebraic-topology homotopy-theory
asked Dec 6 '18 at 15:14
GreggGregg
9318
asked Dec 6 '18 at 15:14
GreggGregg
9318
edited Dec 6 '18 at 15:38
Gregg
asked Dec 6 '18 at 15:14
GreggGregg
9318
asked Dec 6 '18 at 15:14
GreggGregg
9318
asked Dec 6 '18 at 15:14
GreggGregg
9318
9318
$begingroup$
Interesting question. You certainly know Neil Stricklands's paper "The category of CGWH spaces" neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf. It is easy to see that the product of weakly Hausdorff spaces it weakly Hausdorff. Hence by Strickland 2.6, if $X$ is CGWH, then so is $X times I$. Using Strickland 1.10 we see that if a map $i : A to X$ in CGWH is a cofibration in CG, then it is also one in Top. However, there is a still a gap: If $i$ is a cofibration in CGWH, is it one in CG?
$endgroup$
– Paul Frost
Dec 6 '18 at 16:20
$begingroup$
@PaulFrost Actually, I didn’t know about this paper before. I will have a look, thank you
$endgroup$
– Gregg
Dec 6 '18 at 18:02
$begingroup$
Concerning the last question in my post, the answer is negative. It is known that any Serre fibration between CW complexes is a Hurewicz fibration in CGWH, but it may be not Hurewicz in Top (for a proof and relevant counterexample, see M. Steinberger and J. West. Covering homotopy properties of maps between CW complexes or ANRs. Proc. Amer. Math. Soc. 92(1984), 573-577)
$endgroup$
– Gregg
Dec 6 '18 at 18:06
1
$begingroup$
If $i(A)$ is closed in $X$, then $i$ is a cofibration in Top iff $Z = X times 0 cup i(A) times I$ is a retract of $X times I$. It suffices to know that $i$ is a cofibration in CGWH to show that $Z$ is a retract of $X times I$: $Z$ is closed in $X times I$, hence a CGWH space.
$endgroup$
– Paul Frost
Dec 6 '18 at 19:54
$begingroup$
@PaulFrost Actually, $i$ is a cofibration in Top iff the canonical map $l: M_ito Xtimes I$ has a retract iff $Z$ is a retract of $Xtimes I$, so the closedness assumption is not necessary. (However, it can be shown that a cofibration in CGWH is always a closed embedding.) But knowing that a map is cofibration in CGWH is indeed enough to conlude that $l$ has a retract (because the pushout of a diagram in which at least one of the maps is a closed embedding is in CGWH). Thank you!
$endgroup$
– Gregg
Dec 8 '18 at 19:40
add a comment |
$begingroup$
Interesting question. You certainly know Neil Stricklands's paper "The category of CGWH spaces" neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf. It is easy to see that the product of weakly Hausdorff spaces it weakly Hausdorff. Hence by Strickland 2.6, if $X$ is CGWH, then so is $X times I$. Using Strickland 1.10 we see that if a map $i : A to X$ in CGWH is a cofibration in CG, then it is also one in Top. However, there is a still a gap: If $i$ is a cofibration in CGWH, is it one in CG?
$endgroup$
– Paul Frost
Dec 6 '18 at 16:20
$begingroup$
@PaulFrost Actually, I didn’t know about this paper before. I will have a look, thank you
$endgroup$
– Gregg
Dec 6 '18 at 18:02
$begingroup$
Concerning the last question in my post, the answer is negative. It is known that any Serre fibration between CW complexes is a Hurewicz fibration in CGWH, but it may be not Hurewicz in Top (for a proof and relevant counterexample, see M. Steinberger and J. West. Covering homotopy properties of maps between CW complexes or ANRs. Proc. Amer. Math. Soc. 92(1984), 573-577)
$endgroup$
– Gregg
Dec 6 '18 at 18:06
1
$begingroup$
If $i(A)$ is closed in $X$, then $i$ is a cofibration in Top iff $Z = X times 0 cup i(A) times I$ is a retract of $X times I$. It suffices to know that $i$ is a cofibration in CGWH to show that $Z$ is a retract of $X times I$: $Z$ is closed in $X times I$, hence a CGWH space.
$endgroup$
– Paul Frost
Dec 6 '18 at 19:54
$begingroup$
@PaulFrost Actually, $i$ is a cofibration in Top iff the canonical map $l: M_ito Xtimes I$ has a retract iff $Z$ is a retract of $Xtimes I$, so the closedness assumption is not necessary. (However, it can be shown that a cofibration in CGWH is always a closed embedding.) But knowing that a map is cofibration in CGWH is indeed enough to conlude that $l$ has a retract (because the pushout of a diagram in which at least one of the maps is a closed embedding is in CGWH). Thank you!
$endgroup$
– Gregg
Dec 8 '18 at 19:40
$begingroup$
Interesting question. You certainly know Neil Stricklands's paper "The category of CGWH spaces" neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf. It is easy to see that the product of weakly Hausdorff spaces it weakly Hausdorff. Hence by Strickland 2.6, if $X$ is CGWH, then so is $X times I$. Using Strickland 1.10 we see that if a map $i : A to X$ in CGWH is a cofibration in CG, then it is also one in Top. However, there is a still a gap: If $i$ is a cofibration in CGWH, is it one in CG?
$endgroup$
– Paul Frost
Dec 6 '18 at 16:20
$begingroup$
Interesting question. You certainly know Neil Stricklands's paper "The category of CGWH spaces" neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf. It is easy to see that the product of weakly Hausdorff spaces it weakly Hausdorff. Hence by Strickland 2.6, if $X$ is CGWH, then so is $X times I$. Using Strickland 1.10 we see that if a map $i : A to X$ in CGWH is a cofibration in CG, then it is also one in Top. However, there is a still a gap: If $i$ is a cofibration in CGWH, is it one in CG?
$endgroup$
– Paul Frost
Dec 6 '18 at 16:20
$begingroup$
@PaulFrost Actually, I didn’t know about this paper before. I will have a look, thank you
$endgroup$
– Gregg
Dec 6 '18 at 18:02
$begingroup$
@PaulFrost Actually, I didn’t know about this paper before. I will have a look, thank you
$endgroup$
– Gregg
Dec 6 '18 at 18:02
$begingroup$
Concerning the last question in my post, the answer is negative. It is known that any Serre fibration between CW complexes is a Hurewicz fibration in CGWH, but it may be not Hurewicz in Top (for a proof and relevant counterexample, see M. Steinberger and J. West. Covering homotopy properties of maps between CW complexes or ANRs. Proc. Amer. Math. Soc. 92(1984), 573-577)
$endgroup$
– Gregg
Dec 6 '18 at 18:06
$begingroup$
Concerning the last question in my post, the answer is negative. It is known that any Serre fibration between CW complexes is a Hurewicz fibration in CGWH, but it may be not Hurewicz in Top (for a proof and relevant counterexample, see M. Steinberger and J. West. Covering homotopy properties of maps between CW complexes or ANRs. Proc. Amer. Math. Soc. 92(1984), 573-577)
$endgroup$
– Gregg
Dec 6 '18 at 18:06
1
1
$begingroup$
If $i(A)$ is closed in $X$, then $i$ is a cofibration in Top iff $Z = X times 0 cup i(A) times I$ is a retract of $X times I$. It suffices to know that $i$ is a cofibration in CGWH to show that $Z$ is a retract of $X times I$: $Z$ is closed in $X times I$, hence a CGWH space.
$endgroup$
– Paul Frost
Dec 6 '18 at 19:54
$begingroup$
If $i(A)$ is closed in $X$, then $i$ is a cofibration in Top iff $Z = X times 0 cup i(A) times I$ is a retract of $X times I$. It suffices to know that $i$ is a cofibration in CGWH to show that $Z$ is a retract of $X times I$: $Z$ is closed in $X times I$, hence a CGWH space.
$endgroup$
– Paul Frost
Dec 6 '18 at 19:54
$begingroup$
@PaulFrost Actually, $i$ is a cofibration in Top iff the canonical map $l: M_ito Xtimes I$ has a retract iff $Z$ is a retract of $Xtimes I$, so the closedness assumption is not necessary. (However, it can be shown that a cofibration in CGWH is always a closed embedding.) But knowing that a map is cofibration in CGWH is indeed enough to conlude that $l$ has a retract (because the pushout of a diagram in which at least one of the maps is a closed embedding is in CGWH). Thank you!
$endgroup$
– Gregg
Dec 8 '18 at 19:40
$begingroup$
@PaulFrost Actually, $i$ is a cofibration in Top iff the canonical map $l: M_ito Xtimes I$ has a retract iff $Z$ is a retract of $Xtimes I$, so the closedness assumption is not necessary. (However, it can be shown that a cofibration in CGWH is always a closed embedding.) But knowing that a map is cofibration in CGWH is indeed enough to conlude that $l$ has a retract (because the pushout of a diagram in which at least one of the maps is a closed embedding is in CGWH). Thank you!
$endgroup$
– Gregg
Dec 8 '18 at 19:40
add a comment |
1 Answer
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$begingroup$
A map $i: Ato X$ is a cofibration in Top iff the canonical map $l: M_ito Xtimes I$ has a retract iff $Z=Xtimes {0}cup i(A)times I$ is a retract of $Xtimes I$. Knowing that a map is cofibration in CGWH is enough to conlude that $l$ has a retract, because the pushout of a diagram in which at least one of the maps is a closed embedding is in CGWH.
Concerning the last question, the answer is negative, see the comments.
$endgroup$
add a comment |
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$begingroup$
A map $i: Ato X$ is a cofibration in Top iff the canonical map $l: M_ito Xtimes I$ has a retract iff $Z=Xtimes {0}cup i(A)times I$ is a retract of $Xtimes I$. Knowing that a map is cofibration in CGWH is enough to conlude that $l$ has a retract, because the pushout of a diagram in which at least one of the maps is a closed embedding is in CGWH.
Concerning the last question, the answer is negative, see the comments.
$endgroup$
add a comment |
$begingroup$
A map $i: Ato X$ is a cofibration in Top iff the canonical map $l: M_ito Xtimes I$ has a retract iff $Z=Xtimes {0}cup i(A)times I$ is a retract of $Xtimes I$. Knowing that a map is cofibration in CGWH is enough to conlude that $l$ has a retract, because the pushout of a diagram in which at least one of the maps is a closed embedding is in CGWH.
Concerning the last question, the answer is negative, see the comments.
$endgroup$
add a comment |
$begingroup$
A map $i: Ato X$ is a cofibration in Top iff the canonical map $l: M_ito Xtimes I$ has a retract iff $Z=Xtimes {0}cup i(A)times I$ is a retract of $Xtimes I$. Knowing that a map is cofibration in CGWH is enough to conlude that $l$ has a retract, because the pushout of a diagram in which at least one of the maps is a closed embedding is in CGWH.
Concerning the last question, the answer is negative, see the comments.
$endgroup$
A map $i: Ato X$ is a cofibration in Top iff the canonical map $l: M_ito Xtimes I$ has a retract iff $Z=Xtimes {0}cup i(A)times I$ is a retract of $Xtimes I$. Knowing that a map is cofibration in CGWH is enough to conlude that $l$ has a retract, because the pushout of a diagram in which at least one of the maps is a closed embedding is in CGWH.
Concerning the last question, the answer is negative, see the comments.
answered Dec 8 '18 at 19:48
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Gregg
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$begingroup$
Interesting question. You certainly know Neil Stricklands's paper "The category of CGWH spaces" neil-strickland.staff.shef.ac.uk/courses/homotopy/cgwh.pdf. It is easy to see that the product of weakly Hausdorff spaces it weakly Hausdorff. Hence by Strickland 2.6, if $X$ is CGWH, then so is $X times I$. Using Strickland 1.10 we see that if a map $i : A to X$ in CGWH is a cofibration in CG, then it is also one in Top. However, there is a still a gap: If $i$ is a cofibration in CGWH, is it one in CG?
$endgroup$
– Paul Frost
Dec 6 '18 at 16:20
$begingroup$
@PaulFrost Actually, I didn’t know about this paper before. I will have a look, thank you
$endgroup$
– Gregg
Dec 6 '18 at 18:02
$begingroup$
Concerning the last question in my post, the answer is negative. It is known that any Serre fibration between CW complexes is a Hurewicz fibration in CGWH, but it may be not Hurewicz in Top (for a proof and relevant counterexample, see M. Steinberger and J. West. Covering homotopy properties of maps between CW complexes or ANRs. Proc. Amer. Math. Soc. 92(1984), 573-577)
$endgroup$
– Gregg
Dec 6 '18 at 18:06
1
$begingroup$
If $i(A)$ is closed in $X$, then $i$ is a cofibration in Top iff $Z = X times 0 cup i(A) times I$ is a retract of $X times I$. It suffices to know that $i$ is a cofibration in CGWH to show that $Z$ is a retract of $X times I$: $Z$ is closed in $X times I$, hence a CGWH space.
$endgroup$
– Paul Frost
Dec 6 '18 at 19:54
$begingroup$
@PaulFrost Actually, $i$ is a cofibration in Top iff the canonical map $l: M_ito Xtimes I$ has a retract iff $Z$ is a retract of $Xtimes I$, so the closedness assumption is not necessary. (However, it can be shown that a cofibration in CGWH is always a closed embedding.) But knowing that a map is cofibration in CGWH is indeed enough to conlude that $l$ has a retract (because the pushout of a diagram in which at least one of the maps is a closed embedding is in CGWH). Thank you!
$endgroup$
– Gregg
Dec 8 '18 at 19:40