Higher Topos Theory Theorem 2.2.5.3
$begingroup$
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
edited Dec 26 '18 at 17:22
Earthliŋ
534220
asked Dec 26 '18 at 11:31
Oscar P.Oscar P.
1996
$endgroup$
add a comment |
$begingroup$
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
edited Dec 26 '18 at 17:22
Earthliŋ
534220
asked Dec 26 '18 at 11:31
Oscar P.Oscar P.
1996
$endgroup$
add a comment |
$begingroup$
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
edited Dec 26 '18 at 17:22
Earthliŋ
534220
asked Dec 26 '18 at 11:31
Oscar P.Oscar P.
1996
$endgroup$
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
edited Dec 26 '18 at 17:22
Earthliŋ
534220
asked Dec 26 '18 at 11:31
Oscar P.Oscar P.
1996
edited Dec 26 '18 at 17:22
Earthliŋ
534220
asked Dec 26 '18 at 11:31
Oscar P.Oscar P.
1996
edited Dec 26 '18 at 17:22
Earthliŋ
534220
edited Dec 26 '18 at 17:22
Earthliŋ
534220
edited Dec 26 '18 at 17:22
Earthliŋ
534220
534220
asked Dec 26 '18 at 11:31
Oscar P.Oscar P.
1996
asked Dec 26 '18 at 11:31
Oscar P.Oscar P.
1996
asked Dec 26 '18 at 11:31
Oscar P.Oscar P.
1996
1996
add a comment |
add a comment |
1 Answer
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$begingroup$
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.
answered Dec 26 '18 at 15:19
Tim CampionTim Campion
14.8k355128
$endgroup$
add a comment |
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1 Answer
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active
oldest
votes
1 Answer
1
active
oldest
votes
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votes
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votes
$begingroup$
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.
answered Dec 26 '18 at 15:19
Tim CampionTim Campion
14.8k355128
$endgroup$
add a comment |
$begingroup$
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.
answered Dec 26 '18 at 15:19
Tim CampionTim Campion
14.8k355128
$endgroup$
add a comment |
$begingroup$
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.
answered Dec 26 '18 at 15:19
Tim CampionTim Campion
14.8k355128
$endgroup$
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.
answered Dec 26 '18 at 15:19
Tim CampionTim Campion
14.8k355128
answered Dec 26 '18 at 15:19
Tim CampionTim Campion
14.8k355128
answered Dec 26 '18 at 15:19
Tim CampionTim Campion
14.8k355128
answered Dec 26 '18 at 15:19
Tim CampionTim Campion
14.8k355128
14.8k355128
add a comment |
add a comment |
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