(Co-)fibrations in Top and CGWH












0












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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












  • $begingroup$
    @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


















0












$begingroup$


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







share|cite|improve this question











$endgroup$












  • $begingroup$
    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












  • $begingroup$
    @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
















0












0








0





$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 6 '18 at 15:38







Gregg

















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




















  • $begingroup$
    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












1 Answer
1






active

oldest

votes


















1












$begingroup$

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.






share|cite|improve this answer











$endgroup$













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "69"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });














    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3028613%2fco-fibrations-in-top-and-cgwh%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    1












    $begingroup$

    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.






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      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.






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        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.






        share|cite|improve this answer











        $endgroup$



        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.







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        answered Dec 8 '18 at 19:48


























        community wiki





        Gregg































            draft saved

            draft discarded




















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3028613%2fco-fibrations-in-top-and-cgwh%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Ellipse (mathématiques)

            Image
            Pour les articles homonymes, voir Ellipse. L'ellipse est le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante. Section du cône ou projection du cercle. En géométrie, une ellipse est une courbe plane fermée obtenue par l’intersection d’un cône de révolution avec un plan, à condition que celui-ci coupe l'axe de rotation du cône ou du cylindre : c'est une conique d'excentricité strictement comprise entre 0 et 1. On peut également la définir comme le lieu des points dont la somme des distances à deux points fixes, dits foyers, est constante (sa construction par la méthode du jardinier est très simple). Dans la vie courante, l’ellipse est la forme qu'on perçoit en regardant un cercle en perspective, ou la figure formée par l’ombre d'un disque sur une surface plane. On retrouve aussi, en première approximation [ 1 ] , des ellipses dans les trajectoires des corps célestes (planètes, comètes ou satellites arti...
            Read more

            Quarter-circle Tiles

            Image
            up vote 0 down vote favorite I have $99$ identical square tiles, each with a quarter-circle drawn on it like this: [asy] size(1.5cm); draw(Arc((2,0),1,90,180),red+1); draw((0,0)--(2,0)--(2,2)--(0,2)--(0,0)); [/asy] When I arrange the tiles in a $9times 11$ rectangular grid, each with a random orientation, what is the expected value of the number of full circles I form? I think this problem has to do with finding the chance any given 2x2 square has a circle, but I can't find it. expected-value share | cite | improve this question asked Nov 20 at 15:03 6minecraftninja 1 2 ...
            Read more

            Mont Emei

            Image
            Mont Emei.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Vue du sommet Wanfo du mont Emei Géographie Altitude 3 099  m , Wanfo Coordonnées 29° 31′ 11″ nord, 103° 19′ 57″ est Administration Pays   Chine Province Sichuan Ville-préfecture Leshan Géolocalisation sur la carte : Sichuan Mont Emei Géolocalisation sur la carte : Chine Mont Emei modifier   Paysage panoramique du mont Emei, incluant le paysage panoramique du grand Bouddha de Leshan *   Patrimoine mondial de l'UNESCO Passerelle en bois sur le versant ouest Pays   Chine Subdivision Sichuan Type Mixte Critères (iv) (vi) (x) Superficie 15 400 ha Numéro d’identification 779 Zone géographique Asie et Pacifique  ** Année d’inscription 1996 (20 e session) * Descriptif officiel UNESCO ** Classification géog...
            Read more