Higher Topos Theory Theorem 2.2.5.3












8














The following question is found in the proof of Theorem 2.2.5.3 of HTT but since it can be understood in a more general context I will just ask it without stating the theorem.



We have a trivial Kan fibration of simplicial sets $p : S rightarrow T$ where $T$ is an $infty$-category. We wish to show that for any two vertices $x,y$ of $S$, the induced map of simplicial sets $$Map_{mathfrak{C}[S]}(x,y) rightarrow Map_{mathfrak{C}[T]}(p(x),p(y))$$ is a Kan weak equivalence.
We have two results, the first one is that the map $$lvert Hom^R_T(p(x),p(y)) rvert_{Q^bullet} rightarrow Map_{mathfrak{C}[T]}(p(x),p(y))$$ where $Q^bullet$ is the cosimplicial object defined in 2.2.2 is a Kan weak equivalence. The second one is that for any simplicial set, the map $$pi_X : lvert X rvert_{Q^bullet} rightarrow X$$ also defined in Section 2.2.2 is also a Kan weak equivalence.
Lurie says that thanks to those two results, it is enough to show that the map $$Hom^R_S(x,y) rightarrow Hom^R_T(p(x),p(y))$$ is a Kan weak equivalence.



I was thinking of fitting all this into a commutative diagram and using the two-out-of-three property but I am struggling with it. Is this the right way to look at it or am I missing something?



If someone needs more definitions I'll gladly add them.






ct.category-theory higher-category-theory simplicial-stuff







share|cite|improve this question





























    8














    The following question is found in the proof of Theorem 2.2.5.3 of HTT but since it can be understood in a more general context I will just ask it without stating the theorem.



    We have a trivial Kan fibration of simplicial sets $p : S rightarrow T$ where $T$ is an $infty$-category. We wish to show that for any two vertices $x,y$ of $S$, the induced map of simplicial sets $$Map_{mathfrak{C}[S]}(x,y) rightarrow Map_{mathfrak{C}[T]}(p(x),p(y))$$ is a Kan weak equivalence.
    We have two results, the first one is that the map $$lvert Hom^R_T(p(x),p(y)) rvert_{Q^bullet} rightarrow Map_{mathfrak{C}[T]}(p(x),p(y))$$ where $Q^bullet$ is the cosimplicial object defined in 2.2.2 is a Kan weak equivalence. The second one is that for any simplicial set, the map $$pi_X : lvert X rvert_{Q^bullet} rightarrow X$$ also defined in Section 2.2.2 is also a Kan weak equivalence.
    Lurie says that thanks to those two results, it is enough to show that the map $$Hom^R_S(x,y) rightarrow Hom^R_T(p(x),p(y))$$ is a Kan weak equivalence.



    I was thinking of fitting all this into a commutative diagram and using the two-out-of-three property but I am struggling with it. Is this the right way to look at it or am I missing something?



    If someone needs more definitions I'll gladly add them.






    ct.category-theory higher-category-theory simplicial-stuff







    share|cite|improve this question



























      8












      8








      8







      The following question is found in the proof of Theorem 2.2.5.3 of HTT but since it can be understood in a more general context I will just ask it without stating the theorem.



      We have a trivial Kan fibration of simplicial sets $p : S rightarrow T$ where $T$ is an $infty$-category. We wish to show that for any two vertices $x,y$ of $S$, the induced map of simplicial sets $$Map_{mathfrak{C}[S]}(x,y) rightarrow Map_{mathfrak{C}[T]}(p(x),p(y))$$ is a Kan weak equivalence.
      We have two results, the first one is that the map $$lvert Hom^R_T(p(x),p(y)) rvert_{Q^bullet} rightarrow Map_{mathfrak{C}[T]}(p(x),p(y))$$ where $Q^bullet$ is the cosimplicial object defined in 2.2.2 is a Kan weak equivalence. The second one is that for any simplicial set, the map $$pi_X : lvert X rvert_{Q^bullet} rightarrow X$$ also defined in Section 2.2.2 is also a Kan weak equivalence.
      Lurie says that thanks to those two results, it is enough to show that the map $$Hom^R_S(x,y) rightarrow Hom^R_T(p(x),p(y))$$ is a Kan weak equivalence.



      I was thinking of fitting all this into a commutative diagram and using the two-out-of-three property but I am struggling with it. Is this the right way to look at it or am I missing something?



      If someone needs more definitions I'll gladly add them.






      ct.category-theory higher-category-theory simplicial-stuff







      share|cite|improve this question















      The following question is found in the proof of Theorem 2.2.5.3 of HTT but since it can be understood in a more general context I will just ask it without stating the theorem.



      We have a trivial Kan fibration of simplicial sets $p : S rightarrow T$ where $T$ is an $infty$-category. We wish to show that for any two vertices $x,y$ of $S$, the induced map of simplicial sets $$Map_{mathfrak{C}[S]}(x,y) rightarrow Map_{mathfrak{C}[T]}(p(x),p(y))$$ is a Kan weak equivalence.
      We have two results, the first one is that the map $$lvert Hom^R_T(p(x),p(y)) rvert_{Q^bullet} rightarrow Map_{mathfrak{C}[T]}(p(x),p(y))$$ where $Q^bullet$ is the cosimplicial object defined in 2.2.2 is a Kan weak equivalence. The second one is that for any simplicial set, the map $$pi_X : lvert X rvert_{Q^bullet} rightarrow X$$ also defined in Section 2.2.2 is also a Kan weak equivalence.
      Lurie says that thanks to those two results, it is enough to show that the map $$Hom^R_S(x,y) rightarrow Hom^R_T(p(x),p(y))$$ is a Kan weak equivalence.



      I was thinking of fitting all this into a commutative diagram and using the two-out-of-three property but I am struggling with it. Is this the right way to look at it or am I missing something?



      If someone needs more definitions I'll gladly add them.






      ct.category-theory higher-category-theory simplicial-stuff




      ct.category-theory higher-category-theory simplicial-stuff






      share|cite|improve this question















      share|cite|improve this question













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      edited 5 hours ago









      Earthliŋ

      529219




      529219










      asked 10 hours ago









      Valérian Montessuit

      885




      885






















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          11














          From naturality we have the following commutative diagram:



          $require{AMScd}
          begin{CD}
          Map_{mathfrak C[S]}(x,y) @<sim<< |Hom^R_S(x,y)|_{Q_bullet} @>sim>> Hom^R_S(x,y) \
          @VVV @VVV @VVV\
          Map_{mathfrak C[T]}(px,py) @<sim<< |Hom^R_T(px,py)|_{Q_bullet} @>sim>> Hom^R_T(px,py)
          end{CD}$



          From the above two results the horizontal maps are weak equivalences. So if the right hand vertical map is an equivalence, then by 2/3, so is the middle vertical map, so by 2/3, so is the left hand vertical map.






          share|cite|improve this answer





















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            11














            From naturality we have the following commutative diagram:



            $require{AMScd}
            begin{CD}
            Map_{mathfrak C[S]}(x,y) @<sim<< |Hom^R_S(x,y)|_{Q_bullet} @>sim>> Hom^R_S(x,y) \
            @VVV @VVV @VVV\
            Map_{mathfrak C[T]}(px,py) @<sim<< |Hom^R_T(px,py)|_{Q_bullet} @>sim>> Hom^R_T(px,py)
            end{CD}$



            From the above two results the horizontal maps are weak equivalences. So if the right hand vertical map is an equivalence, then by 2/3, so is the middle vertical map, so by 2/3, so is the left hand vertical map.






            share|cite|improve this answer


























              11














              From naturality we have the following commutative diagram:



              $require{AMScd}
              begin{CD}
              Map_{mathfrak C[S]}(x,y) @<sim<< |Hom^R_S(x,y)|_{Q_bullet} @>sim>> Hom^R_S(x,y) \
              @VVV @VVV @VVV\
              Map_{mathfrak C[T]}(px,py) @<sim<< |Hom^R_T(px,py)|_{Q_bullet} @>sim>> Hom^R_T(px,py)
              end{CD}$



              From the above two results the horizontal maps are weak equivalences. So if the right hand vertical map is an equivalence, then by 2/3, so is the middle vertical map, so by 2/3, so is the left hand vertical map.






              share|cite|improve this answer
























                11












                11








                11






                From naturality we have the following commutative diagram:



                $require{AMScd}
                begin{CD}
                Map_{mathfrak C[S]}(x,y) @<sim<< |Hom^R_S(x,y)|_{Q_bullet} @>sim>> Hom^R_S(x,y) \
                @VVV @VVV @VVV\
                Map_{mathfrak C[T]}(px,py) @<sim<< |Hom^R_T(px,py)|_{Q_bullet} @>sim>> Hom^R_T(px,py)
                end{CD}$



                From the above two results the horizontal maps are weak equivalences. So if the right hand vertical map is an equivalence, then by 2/3, so is the middle vertical map, so by 2/3, so is the left hand vertical map.






                share|cite|improve this answer












                From naturality we have the following commutative diagram:



                $require{AMScd}
                begin{CD}
                Map_{mathfrak C[S]}(x,y) @<sim<< |Hom^R_S(x,y)|_{Q_bullet} @>sim>> Hom^R_S(x,y) \
                @VVV @VVV @VVV\
                Map_{mathfrak C[T]}(px,py) @<sim<< |Hom^R_T(px,py)|_{Q_bullet} @>sim>> Hom^R_T(px,py)
                end{CD}$



                From the above two results the horizontal maps are weak equivalences. So if the right hand vertical map is an equivalence, then by 2/3, so is the middle vertical map, so by 2/3, so is the left hand vertical map.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 7 hours ago









                Tim Campion

                13.3k354122




                13.3k354122






























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