homotopy and (co)filtered limits












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Suppose we have a (co)filtered digaram $dots rightarrow X_{2}rightarrow X_{1}$ of topological space. Is is true that the natural map $pi_{0}[lim X_{i}]rightarrow lim pi_{0}(X_{i})$ is an isomorphism ?






at.algebraic-topology homotopy-theory







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  • 7




    $begingroup$
    No. For example, take all your $X_i$ to be the $1$-sphere, and all the maps the degree $2$ map. The limit is actually not pathconnected.
    $endgroup$
    – Achim Krause
    Dec 6 '18 at 21:20
















11












$begingroup$


Suppose we have a (co)filtered digaram $dots rightarrow X_{2}rightarrow X_{1}$ of topological space. Is is true that the natural map $pi_{0}[lim X_{i}]rightarrow lim pi_{0}(X_{i})$ is an isomorphism ?






at.algebraic-topology homotopy-theory







share|cite|improve this question











$endgroup$








  • 7




    $begingroup$
    No. For example, take all your $X_i$ to be the $1$-sphere, and all the maps the degree $2$ map. The limit is actually not pathconnected.
    $endgroup$
    – Achim Krause
    Dec 6 '18 at 21:20














11












11








11


2



$begingroup$


Suppose we have a (co)filtered digaram $dots rightarrow X_{2}rightarrow X_{1}$ of topological space. Is is true that the natural map $pi_{0}[lim X_{i}]rightarrow lim pi_{0}(X_{i})$ is an isomorphism ?






at.algebraic-topology homotopy-theory







share|cite|improve this question











$endgroup$




Suppose we have a (co)filtered digaram $dots rightarrow X_{2}rightarrow X_{1}$ of topological space. Is is true that the natural map $pi_{0}[lim X_{i}]rightarrow lim pi_{0}(X_{i})$ is an isomorphism ?






at.algebraic-topology homotopy-theory




at.algebraic-topology homotopy-theory






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edited Dec 7 '18 at 14:58









Rad80

1032




1032










asked Dec 6 '18 at 20:23









OfraOfra

601620




601620








  • 7




    $begingroup$
    No. For example, take all your $X_i$ to be the $1$-sphere, and all the maps the degree $2$ map. The limit is actually not pathconnected.
    $endgroup$
    – Achim Krause
    Dec 6 '18 at 21:20














  • 7




    $begingroup$
    No. For example, take all your $X_i$ to be the $1$-sphere, and all the maps the degree $2$ map. The limit is actually not pathconnected.
    $endgroup$
    – Achim Krause
    Dec 6 '18 at 21:20








7




7




$begingroup$
No. For example, take all your $X_i$ to be the $1$-sphere, and all the maps the degree $2$ map. The limit is actually not pathconnected.
$endgroup$
– Achim Krause
Dec 6 '18 at 21:20




$begingroup$
No. For example, take all your $X_i$ to be the $1$-sphere, and all the maps the degree $2$ map. The limit is actually not pathconnected.
$endgroup$
– Achim Krause
Dec 6 '18 at 21:20










1 Answer
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29












$begingroup$

This is not true, for two distinct reasons.




  1. The first is that the inverse system of spaces may not behave well homotopy-theoretically. If $X_n = [n, infty) subset Bbb R$, then the limit of $dots to X_2 to X_1 to X_0$ is empty. However, on path components it is the constant system $dots to * to * to *$, with limit $*$. Roughly, you might have a path component that is represented by any space $X_i$ that is not represented by any compatible system of points. This might make $pi_0 lim X_i to lim pi_0 X_i$ not surjective.


  2. The second is the opposite: the map $pi_0 lim X_i to lim pi_0 X_i$ may not be injective. In this case, you may have two points $x$ and $y$ in the limit such that the images in any individual $x_i$ are connected by a path, but where no path can be compatibly lifted all the way up the tower. For example, if $f:S^1 to S^1$ is a degree-2 covering map, then the limit of the tower $$dots xrightarrow{f} S^1 xrightarrow{f} S^1 xrightarrow{f} S^1$$ is called the 2-adic solenoid and it has uncountably many path components.



The first problem goes away if the maps $X_i to X_{i-1}$ are fibrations, and in this case we often call the limit a homotopy limit. The second problem does not go away in this case, but Milnor proved that $pi_0 lim X_i$ is built out of two terms: $lim(pi_0 X_i)$ and a second term called $lim^1(pi_1 X_i)$. In particular, if the spaces $X_i$ are simply-connected then there are no contributions from the second term, and so there will be an isomorphism $pi_0(lim X_i) to lim pi_0(X_i)$.






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    29












    $begingroup$

    This is not true, for two distinct reasons.




    1. The first is that the inverse system of spaces may not behave well homotopy-theoretically. If $X_n = [n, infty) subset Bbb R$, then the limit of $dots to X_2 to X_1 to X_0$ is empty. However, on path components it is the constant system $dots to * to * to *$, with limit $*$. Roughly, you might have a path component that is represented by any space $X_i$ that is not represented by any compatible system of points. This might make $pi_0 lim X_i to lim pi_0 X_i$ not surjective.


    2. The second is the opposite: the map $pi_0 lim X_i to lim pi_0 X_i$ may not be injective. In this case, you may have two points $x$ and $y$ in the limit such that the images in any individual $x_i$ are connected by a path, but where no path can be compatibly lifted all the way up the tower. For example, if $f:S^1 to S^1$ is a degree-2 covering map, then the limit of the tower $$dots xrightarrow{f} S^1 xrightarrow{f} S^1 xrightarrow{f} S^1$$ is called the 2-adic solenoid and it has uncountably many path components.



    The first problem goes away if the maps $X_i to X_{i-1}$ are fibrations, and in this case we often call the limit a homotopy limit. The second problem does not go away in this case, but Milnor proved that $pi_0 lim X_i$ is built out of two terms: $lim(pi_0 X_i)$ and a second term called $lim^1(pi_1 X_i)$. In particular, if the spaces $X_i$ are simply-connected then there are no contributions from the second term, and so there will be an isomorphism $pi_0(lim X_i) to lim pi_0(X_i)$.






    share|cite|improve this answer









    $endgroup$


















      29












      $begingroup$

      This is not true, for two distinct reasons.




      1. The first is that the inverse system of spaces may not behave well homotopy-theoretically. If $X_n = [n, infty) subset Bbb R$, then the limit of $dots to X_2 to X_1 to X_0$ is empty. However, on path components it is the constant system $dots to * to * to *$, with limit $*$. Roughly, you might have a path component that is represented by any space $X_i$ that is not represented by any compatible system of points. This might make $pi_0 lim X_i to lim pi_0 X_i$ not surjective.


      2. The second is the opposite: the map $pi_0 lim X_i to lim pi_0 X_i$ may not be injective. In this case, you may have two points $x$ and $y$ in the limit such that the images in any individual $x_i$ are connected by a path, but where no path can be compatibly lifted all the way up the tower. For example, if $f:S^1 to S^1$ is a degree-2 covering map, then the limit of the tower $$dots xrightarrow{f} S^1 xrightarrow{f} S^1 xrightarrow{f} S^1$$ is called the 2-adic solenoid and it has uncountably many path components.



      The first problem goes away if the maps $X_i to X_{i-1}$ are fibrations, and in this case we often call the limit a homotopy limit. The second problem does not go away in this case, but Milnor proved that $pi_0 lim X_i$ is built out of two terms: $lim(pi_0 X_i)$ and a second term called $lim^1(pi_1 X_i)$. In particular, if the spaces $X_i$ are simply-connected then there are no contributions from the second term, and so there will be an isomorphism $pi_0(lim X_i) to lim pi_0(X_i)$.






      share|cite|improve this answer









      $endgroup$
















        29












        29








        29





        $begingroup$

        This is not true, for two distinct reasons.




        1. The first is that the inverse system of spaces may not behave well homotopy-theoretically. If $X_n = [n, infty) subset Bbb R$, then the limit of $dots to X_2 to X_1 to X_0$ is empty. However, on path components it is the constant system $dots to * to * to *$, with limit $*$. Roughly, you might have a path component that is represented by any space $X_i$ that is not represented by any compatible system of points. This might make $pi_0 lim X_i to lim pi_0 X_i$ not surjective.


        2. The second is the opposite: the map $pi_0 lim X_i to lim pi_0 X_i$ may not be injective. In this case, you may have two points $x$ and $y$ in the limit such that the images in any individual $x_i$ are connected by a path, but where no path can be compatibly lifted all the way up the tower. For example, if $f:S^1 to S^1$ is a degree-2 covering map, then the limit of the tower $$dots xrightarrow{f} S^1 xrightarrow{f} S^1 xrightarrow{f} S^1$$ is called the 2-adic solenoid and it has uncountably many path components.



        The first problem goes away if the maps $X_i to X_{i-1}$ are fibrations, and in this case we often call the limit a homotopy limit. The second problem does not go away in this case, but Milnor proved that $pi_0 lim X_i$ is built out of two terms: $lim(pi_0 X_i)$ and a second term called $lim^1(pi_1 X_i)$. In particular, if the spaces $X_i$ are simply-connected then there are no contributions from the second term, and so there will be an isomorphism $pi_0(lim X_i) to lim pi_0(X_i)$.






        share|cite|improve this answer









        $endgroup$



        This is not true, for two distinct reasons.




        1. The first is that the inverse system of spaces may not behave well homotopy-theoretically. If $X_n = [n, infty) subset Bbb R$, then the limit of $dots to X_2 to X_1 to X_0$ is empty. However, on path components it is the constant system $dots to * to * to *$, with limit $*$. Roughly, you might have a path component that is represented by any space $X_i$ that is not represented by any compatible system of points. This might make $pi_0 lim X_i to lim pi_0 X_i$ not surjective.


        2. The second is the opposite: the map $pi_0 lim X_i to lim pi_0 X_i$ may not be injective. In this case, you may have two points $x$ and $y$ in the limit such that the images in any individual $x_i$ are connected by a path, but where no path can be compatibly lifted all the way up the tower. For example, if $f:S^1 to S^1$ is a degree-2 covering map, then the limit of the tower $$dots xrightarrow{f} S^1 xrightarrow{f} S^1 xrightarrow{f} S^1$$ is called the 2-adic solenoid and it has uncountably many path components.



        The first problem goes away if the maps $X_i to X_{i-1}$ are fibrations, and in this case we often call the limit a homotopy limit. The second problem does not go away in this case, but Milnor proved that $pi_0 lim X_i$ is built out of two terms: $lim(pi_0 X_i)$ and a second term called $lim^1(pi_1 X_i)$. In particular, if the spaces $X_i$ are simply-connected then there are no contributions from the second term, and so there will be an isomorphism $pi_0(lim X_i) to lim pi_0(X_i)$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 6 '18 at 21:26









        Tyler LawsonTyler Lawson

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