Simplicial set represented by an (unordered) set












1












$begingroup$


Let $X$ be a (finite if you want) set and form the simplicial set $F^{bullet}(X)$ with
$$
F^{n}(X) = mathrm{Hom}_{mathrm{set}} ([n], X)
$$
where the right hand side denotes arbitrary maps of sets (of course
it wouldn't make sense to say order preserving as $X$ doesn't come
with an order).



I'm wondering about a description of $F^{bullet}(X)$. For example if $X = {0,1}$ then there are 2 0-simplices, may as well call them $[0] and [1]$ and 2 1-simplices $[0, 1]$ and $[1,0]$ glued together to form a copy of $S^1$.



Edit: as pointed out in Goodwillie's answer, this is not the end of the story, there are way more higher dimensional non-degenerate simplices.



Is there an analogous description when $X = {0, 1, 2}$?



A closely related question is whether there's a right adjoint to the
forgetful functor from the simplex category $Delta$ (finite ordered
sets) to, say, finite (unordered) sets -- and if so what is it.



Example where such simplicial sets arise: given a map of topological spaces $f: X
to Y$ we can always form a
simplicial object $mathcal{S}^{bullet}(f)$ with
$$
mathcal{S}^{n} = prodnolimits_{X}^{n} = underbrace{X times_{Y}
cdots times_{Y} X}_{ntext{ times }}
$$
with face and degeneracy maps given by projections and diagonals
respectively. Taking connected components gives a simplicial set.



When $Y$ is the union $bigcup_{i=1}^{N} H_{i}$ of the coordinate
hyperplanes in $mathbb{C}^{N}$ and $f: X=coprod_{i=1}^{N} H_{i} to
bigcup_{i=1}^{N} H_{i}=Y$ is the obvious map, I believe the simplicial
set we get is $F^{bullet}({1, dots, n})$.






ag.algebraic-geometry at.algebraic-topology ct.category-theory simplicial-stuff hyperplane-arrangements







share|cite|improve this question











$endgroup$

















    1












    $begingroup$


    Let $X$ be a (finite if you want) set and form the simplicial set $F^{bullet}(X)$ with
    $$
    F^{n}(X) = mathrm{Hom}_{mathrm{set}} ([n], X)
    $$
    where the right hand side denotes arbitrary maps of sets (of course
    it wouldn't make sense to say order preserving as $X$ doesn't come
    with an order).



    I'm wondering about a description of $F^{bullet}(X)$. For example if $X = {0,1}$ then there are 2 0-simplices, may as well call them $[0] and [1]$ and 2 1-simplices $[0, 1]$ and $[1,0]$ glued together to form a copy of $S^1$.



    Edit: as pointed out in Goodwillie's answer, this is not the end of the story, there are way more higher dimensional non-degenerate simplices.



    Is there an analogous description when $X = {0, 1, 2}$?



    A closely related question is whether there's a right adjoint to the
    forgetful functor from the simplex category $Delta$ (finite ordered
    sets) to, say, finite (unordered) sets -- and if so what is it.



    Example where such simplicial sets arise: given a map of topological spaces $f: X
    to Y$     we can always form a
    simplicial object $mathcal{S}^{bullet}(f)$ with
    $$
    mathcal{S}^{n} = prodnolimits_{X}^{n} = underbrace{X times_{Y}
    cdots times_{Y} X}_{ntext{ times }}
    $$
    with face and degeneracy maps given by projections and diagonals
    respectively. Taking connected components gives a simplicial set.



    When $Y$ is the union $bigcup_{i=1}^{N} H_{i}$ of the coordinate
    hyperplanes in $mathbb{C}^{N}$ and $f: X=coprod_{i=1}^{N} H_{i} to
    bigcup_{i=1}^{N} H_{i}=Y$     is the obvious map, I believe the simplicial
    set we get is $F^{bullet}({1, dots, n})$.






    ag.algebraic-geometry at.algebraic-topology ct.category-theory simplicial-stuff hyperplane-arrangements







    share|cite|improve this question











    $endgroup$















      1












      1








      1





      $begingroup$


      Let $X$ be a (finite if you want) set and form the simplicial set $F^{bullet}(X)$ with
      $$
      F^{n}(X) = mathrm{Hom}_{mathrm{set}} ([n], X)
      $$
      where the right hand side denotes arbitrary maps of sets (of course
      it wouldn't make sense to say order preserving as $X$ doesn't come
      with an order).



      I'm wondering about a description of $F^{bullet}(X)$. For example if $X = {0,1}$ then there are 2 0-simplices, may as well call them $[0] and [1]$ and 2 1-simplices $[0, 1]$ and $[1,0]$ glued together to form a copy of $S^1$.



      Edit: as pointed out in Goodwillie's answer, this is not the end of the story, there are way more higher dimensional non-degenerate simplices.



      Is there an analogous description when $X = {0, 1, 2}$?



      A closely related question is whether there's a right adjoint to the
      forgetful functor from the simplex category $Delta$ (finite ordered
      sets) to, say, finite (unordered) sets -- and if so what is it.



      Example where such simplicial sets arise: given a map of topological spaces $f: X
      to Y$       we can always form a
      simplicial object $mathcal{S}^{bullet}(f)$ with
      $$
      mathcal{S}^{n} = prodnolimits_{X}^{n} = underbrace{X times_{Y}
      cdots times_{Y} X}_{ntext{ times }}
      $$
      with face and degeneracy maps given by projections and diagonals
      respectively. Taking connected components gives a simplicial set.



      When $Y$ is the union $bigcup_{i=1}^{N} H_{i}$ of the coordinate
      hyperplanes in $mathbb{C}^{N}$ and $f: X=coprod_{i=1}^{N} H_{i} to
      bigcup_{i=1}^{N} H_{i}=Y$       is the obvious map, I believe the simplicial
      set we get is $F^{bullet}({1, dots, n})$.






      ag.algebraic-geometry at.algebraic-topology ct.category-theory simplicial-stuff hyperplane-arrangements







      share|cite|improve this question











      $endgroup$




      Let $X$ be a (finite if you want) set and form the simplicial set $F^{bullet}(X)$ with
      $$
      F^{n}(X) = mathrm{Hom}_{mathrm{set}} ([n], X)
      $$
      where the right hand side denotes arbitrary maps of sets (of course
      it wouldn't make sense to say order preserving as $X$ doesn't come
      with an order).



      I'm wondering about a description of $F^{bullet}(X)$. For example if $X = {0,1}$ then there are 2 0-simplices, may as well call them $[0] and [1]$ and 2 1-simplices $[0, 1]$ and $[1,0]$ glued together to form a copy of $S^1$.



      Edit: as pointed out in Goodwillie's answer, this is not the end of the story, there are way more higher dimensional non-degenerate simplices.



      Is there an analogous description when $X = {0, 1, 2}$?



      A closely related question is whether there's a right adjoint to the
      forgetful functor from the simplex category $Delta$ (finite ordered
      sets) to, say, finite (unordered) sets -- and if so what is it.



      Example where such simplicial sets arise: given a map of topological spaces $f: X
      to Y$       we can always form a
      simplicial object $mathcal{S}^{bullet}(f)$ with
      $$
      mathcal{S}^{n} = prodnolimits_{X}^{n} = underbrace{X times_{Y}
      cdots times_{Y} X}_{ntext{ times }}
      $$
      with face and degeneracy maps given by projections and diagonals
      respectively. Taking connected components gives a simplicial set.



      When $Y$ is the union $bigcup_{i=1}^{N} H_{i}$ of the coordinate
      hyperplanes in $mathbb{C}^{N}$ and $f: X=coprod_{i=1}^{N} H_{i} to
      bigcup_{i=1}^{N} H_{i}=Y$       is the obvious map, I believe the simplicial
      set we get is $F^{bullet}({1, dots, n})$.






      ag.algebraic-geometry at.algebraic-topology ct.category-theory simplicial-stuff hyperplane-arrangements




      ag.algebraic-geometry at.algebraic-topology ct.category-theory simplicial-stuff hyperplane-arrangements






      share|cite|improve this question















      share|cite|improve this question













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      edited 4 hours ago







      cgodfrey

















      asked 5 hours ago









      cgodfreycgodfrey

      35819




      35819






















          1 Answer
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          $begingroup$

          You are overlooking some nondegenerate simplices. For example, when $X={0,1}$ there are the $2$-cells $[0,1,0]$ and $[1,0,1]$. In fact, the thing you call $F^bullet(X)$ is infinite dimensional if $X$ has more than one element.



          It is contractible whenever $X$ is non-empty; this can be seen by identifying it with the nerve of a category, a category equivalent to the point category with one morphism.



          If $X=G$ has a group structure then $F^bullet(G)$ is often called $EG$; it is a contractible space with free $G$-action.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Ah! Absolutely overlooked those, many thanks. Quick follow up: as an intermediate step, can we view $F^bullet(X)$ as the nerve of the category with objects the points $x in X$ and with a unique morphism $x to y$ for every 2 points $x, y in X$ (this is at least the case when $X= G$ a group and we build $EG$ as the nerve of the Cayley graph I think...)? Then including a point $x in X$ with its identity morphism would give an equivalence of categories ...
            $endgroup$
            – cgodfrey
            4 hours ago












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          $begingroup$

          You are overlooking some nondegenerate simplices. For example, when $X={0,1}$ there are the $2$-cells $[0,1,0]$ and $[1,0,1]$. In fact, the thing you call $F^bullet(X)$ is infinite dimensional if $X$ has more than one element.



          It is contractible whenever $X$ is non-empty; this can be seen by identifying it with the nerve of a category, a category equivalent to the point category with one morphism.



          If $X=G$ has a group structure then $F^bullet(G)$ is often called $EG$; it is a contractible space with free $G$-action.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Ah! Absolutely overlooked those, many thanks. Quick follow up: as an intermediate step, can we view $F^bullet(X)$ as the nerve of the category with objects the points $x in X$ and with a unique morphism $x to y$ for every 2 points $x, y in X$ (this is at least the case when $X= G$ a group and we build $EG$ as the nerve of the Cayley graph I think...)? Then including a point $x in X$ with its identity morphism would give an equivalence of categories ...
            $endgroup$
            – cgodfrey
            4 hours ago
















          2












          $begingroup$

          You are overlooking some nondegenerate simplices. For example, when $X={0,1}$ there are the $2$-cells $[0,1,0]$ and $[1,0,1]$. In fact, the thing you call $F^bullet(X)$ is infinite dimensional if $X$ has more than one element.



          It is contractible whenever $X$ is non-empty; this can be seen by identifying it with the nerve of a category, a category equivalent to the point category with one morphism.



          If $X=G$ has a group structure then $F^bullet(G)$ is often called $EG$; it is a contractible space with free $G$-action.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Ah! Absolutely overlooked those, many thanks. Quick follow up: as an intermediate step, can we view $F^bullet(X)$ as the nerve of the category with objects the points $x in X$ and with a unique morphism $x to y$ for every 2 points $x, y in X$ (this is at least the case when $X= G$ a group and we build $EG$ as the nerve of the Cayley graph I think...)? Then including a point $x in X$ with its identity morphism would give an equivalence of categories ...
            $endgroup$
            – cgodfrey
            4 hours ago














          2












          2








          2





          $begingroup$

          You are overlooking some nondegenerate simplices. For example, when $X={0,1}$ there are the $2$-cells $[0,1,0]$ and $[1,0,1]$. In fact, the thing you call $F^bullet(X)$ is infinite dimensional if $X$ has more than one element.



          It is contractible whenever $X$ is non-empty; this can be seen by identifying it with the nerve of a category, a category equivalent to the point category with one morphism.



          If $X=G$ has a group structure then $F^bullet(G)$ is often called $EG$; it is a contractible space with free $G$-action.






          share|cite|improve this answer









          $endgroup$



          You are overlooking some nondegenerate simplices. For example, when $X={0,1}$ there are the $2$-cells $[0,1,0]$ and $[1,0,1]$. In fact, the thing you call $F^bullet(X)$ is infinite dimensional if $X$ has more than one element.



          It is contractible whenever $X$ is non-empty; this can be seen by identifying it with the nerve of a category, a category equivalent to the point category with one morphism.



          If $X=G$ has a group structure then $F^bullet(G)$ is often called $EG$; it is a contractible space with free $G$-action.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered 4 hours ago









          Tom GoodwillieTom Goodwillie

          40.5k3111201




          40.5k3111201












          • $begingroup$
            Ah! Absolutely overlooked those, many thanks. Quick follow up: as an intermediate step, can we view $F^bullet(X)$ as the nerve of the category with objects the points $x in X$ and with a unique morphism $x to y$ for every 2 points $x, y in X$ (this is at least the case when $X= G$ a group and we build $EG$ as the nerve of the Cayley graph I think...)? Then including a point $x in X$ with its identity morphism would give an equivalence of categories ...
            $endgroup$
            – cgodfrey
            4 hours ago


















          • $begingroup$
            Ah! Absolutely overlooked those, many thanks. Quick follow up: as an intermediate step, can we view $F^bullet(X)$ as the nerve of the category with objects the points $x in X$ and with a unique morphism $x to y$ for every 2 points $x, y in X$ (this is at least the case when $X= G$ a group and we build $EG$ as the nerve of the Cayley graph I think...)? Then including a point $x in X$ with its identity morphism would give an equivalence of categories ...
            $endgroup$
            – cgodfrey
            4 hours ago
















          $begingroup$
          Ah! Absolutely overlooked those, many thanks. Quick follow up: as an intermediate step, can we view $F^bullet(X)$ as the nerve of the category with objects the points $x in X$ and with a unique morphism $x to y$ for every 2 points $x, y in X$ (this is at least the case when $X= G$ a group and we build $EG$ as the nerve of the Cayley graph I think...)? Then including a point $x in X$ with its identity morphism would give an equivalence of categories ...
          $endgroup$
          – cgodfrey
          4 hours ago




          $begingroup$
          Ah! Absolutely overlooked those, many thanks. Quick follow up: as an intermediate step, can we view $F^bullet(X)$ as the nerve of the category with objects the points $x in X$ and with a unique morphism $x to y$ for every 2 points $x, y in X$ (this is at least the case when $X= G$ a group and we build $EG$ as the nerve of the Cayley graph I think...)? Then including a point $x in X$ with its identity morphism would give an equivalence of categories ...
          $endgroup$
          – cgodfrey
          4 hours ago


















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