Need help understanding comment in Higher Topos Theory
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I am having trouble understanding this part in Lurie's Higher Topos Theory. This can be found in section 2.4.4.4 right after Lemma 2.4.4.1.
Lemma 2.4.4.1. Let$ p : mathcal{C} rightarrow mathcal{D}$ be an inner fibration of $infty$-categories and
let $X, Y in mathcal{C}$. The induced map
$$phi : Hom^R_{mathcal{C}}(X,Y) rightarrow Hom^R_{mathcal{C}}(p(X),p(Y))$$ is a
Kan fibration.
[...]
Suppose the coindtions of Lemma 2.4.4.1 are satisfied. Let us consider the problem of computing the fiber of $phi$ over a vertex $overline{e} : p(X) rightarrow p(Y)$ of $Hom^R_{mathcal{C}}(p(X), p(Y))$. Suppose that there is a $p$-Cartesian edge $e : X' rightarrow Y$ lifting $overline{e}$. By definition, we have a trivial fibration
$$psi : mathcal{C}_{/e} rightarrow mathcal{C}_{/y} times_{}mathcal{D}_{/p(y)} mathcal{D}_{/overline{e}}.$$
Consider the $2$-simplex $sigma = s_1(overline{e})$ regarded as a vertex of $mathcal{D}_{/overline{e}}$. Passing to the fiber, we obtain a trivial fibration
$$ F rightarrow phi^{-1}(overline{e}),$$ where $F$ denotes the fiber of $mathcal{C}_{/e} rightarrow mathcal{D}_{/overline{e}} times_{mathcal{D}_{/p(x)}} mathcal{C}$ over the point $(sigma, x)$.
I am not understanding exactly what he means by "taking fibers". I see that both $F$ and $phi^{-1}(overline{e})$ are fibers, but I don't know how he obtains the trivial fibration between them. I was thinking that both those fibers are defined by pullbacks which are actually homotopy pullbacks and maybe we can find a map of diagram in which all component are weak equivalences to have that those two fibers are weakly equivalent. However I can't find those map and it would not show the that this map is a fibration.
simplicial-stuff higher-category-theory
asked Jan 8 at 18:13
Oscar P.Oscar P.
766
$endgroup$
add a comment |
$begingroup$
I am having trouble understanding this part in Lurie's Higher Topos Theory. This can be found in section 2.4.4.4 right after Lemma 2.4.4.1.
Lemma 2.4.4.1. Let$ p : mathcal{C} rightarrow mathcal{D}$ be an inner fibration of $infty$-categories and
let $X, Y in mathcal{C}$. The induced map
$$phi : Hom^R_{mathcal{C}}(X,Y) rightarrow Hom^R_{mathcal{C}}(p(X),p(Y))$$ is a
Kan fibration.
[...]
Suppose the coindtions of Lemma 2.4.4.1 are satisfied. Let us consider the problem of computing the fiber of $phi$ over a vertex $overline{e} : p(X) rightarrow p(Y)$ of $Hom^R_{mathcal{C}}(p(X), p(Y))$. Suppose that there is a $p$-Cartesian edge $e : X' rightarrow Y$ lifting $overline{e}$. By definition, we have a trivial fibration
$$psi : mathcal{C}_{/e} rightarrow mathcal{C}_{/y} times_{}mathcal{D}_{/p(y)} mathcal{D}_{/overline{e}}.$$
Consider the $2$-simplex $sigma = s_1(overline{e})$ regarded as a vertex of $mathcal{D}_{/overline{e}}$. Passing to the fiber, we obtain a trivial fibration
$$ F rightarrow phi^{-1}(overline{e}),$$ where $F$ denotes the fiber of $mathcal{C}_{/e} rightarrow mathcal{D}_{/overline{e}} times_{mathcal{D}_{/p(x)}} mathcal{C}$ over the point $(sigma, x)$.
I am not understanding exactly what he means by "taking fibers". I see that both $F$ and $phi^{-1}(overline{e})$ are fibers, but I don't know how he obtains the trivial fibration between them. I was thinking that both those fibers are defined by pullbacks which are actually homotopy pullbacks and maybe we can find a map of diagram in which all component are weak equivalences to have that those two fibers are weakly equivalent. However I can't find those map and it would not show the that this map is a fibration.
simplicial-stuff higher-category-theory
asked Jan 8 at 18:13
Oscar P.Oscar P.
766
$endgroup$
add a comment |
$begingroup$
I am having trouble understanding this part in Lurie's Higher Topos Theory. This can be found in section 2.4.4.4 right after Lemma 2.4.4.1.
Lemma 2.4.4.1. Let$ p : mathcal{C} rightarrow mathcal{D}$ be an inner fibration of $infty$-categories and
let $X, Y in mathcal{C}$. The induced map
$$phi : Hom^R_{mathcal{C}}(X,Y) rightarrow Hom^R_{mathcal{C}}(p(X),p(Y))$$ is a
Kan fibration.
[...]
Suppose the coindtions of Lemma 2.4.4.1 are satisfied. Let us consider the problem of computing the fiber of $phi$ over a vertex $overline{e} : p(X) rightarrow p(Y)$ of $Hom^R_{mathcal{C}}(p(X), p(Y))$. Suppose that there is a $p$-Cartesian edge $e : X' rightarrow Y$ lifting $overline{e}$. By definition, we have a trivial fibration
$$psi : mathcal{C}_{/e} rightarrow mathcal{C}_{/y} times_{}mathcal{D}_{/p(y)} mathcal{D}_{/overline{e}}.$$
Consider the $2$-simplex $sigma = s_1(overline{e})$ regarded as a vertex of $mathcal{D}_{/overline{e}}$. Passing to the fiber, we obtain a trivial fibration
$$ F rightarrow phi^{-1}(overline{e}),$$ where $F$ denotes the fiber of $mathcal{C}_{/e} rightarrow mathcal{D}_{/overline{e}} times_{mathcal{D}_{/p(x)}} mathcal{C}$ over the point $(sigma, x)$.
I am not understanding exactly what he means by "taking fibers". I see that both $F$ and $phi^{-1}(overline{e})$ are fibers, but I don't know how he obtains the trivial fibration between them. I was thinking that both those fibers are defined by pullbacks which are actually homotopy pullbacks and maybe we can find a map of diagram in which all component are weak equivalences to have that those two fibers are weakly equivalent. However I can't find those map and it would not show the that this map is a fibration.
simplicial-stuff higher-category-theory
asked Jan 8 at 18:13
Oscar P.Oscar P.
766
$endgroup$
I am having trouble understanding this part in Lurie's Higher Topos Theory. This can be found in section 2.4.4.4 right after Lemma 2.4.4.1.
Lemma 2.4.4.1. Let$ p : mathcal{C} rightarrow mathcal{D}$ be an inner fibration of $infty$-categories and
let $X, Y in mathcal{C}$. The induced map
$$phi : Hom^R_{mathcal{C}}(X,Y) rightarrow Hom^R_{mathcal{C}}(p(X),p(Y))$$ is a
Kan fibration.
[...]
Suppose the coindtions of Lemma 2.4.4.1 are satisfied. Let us consider the problem of computing the fiber of $phi$ over a vertex $overline{e} : p(X) rightarrow p(Y)$ of $Hom^R_{mathcal{C}}(p(X), p(Y))$. Suppose that there is a $p$-Cartesian edge $e : X' rightarrow Y$ lifting $overline{e}$. By definition, we have a trivial fibration
$$psi : mathcal{C}_{/e} rightarrow mathcal{C}_{/y} times_{}mathcal{D}_{/p(y)} mathcal{D}_{/overline{e}}.$$
Consider the $2$-simplex $sigma = s_1(overline{e})$ regarded as a vertex of $mathcal{D}_{/overline{e}}$. Passing to the fiber, we obtain a trivial fibration
$$ F rightarrow phi^{-1}(overline{e}),$$ where $F$ denotes the fiber of $mathcal{C}_{/e} rightarrow mathcal{D}_{/overline{e}} times_{mathcal{D}_{/p(x)}} mathcal{C}$ over the point $(sigma, x)$.
I am not understanding exactly what he means by "taking fibers". I see that both $F$ and $phi^{-1}(overline{e})$ are fibers, but I don't know how he obtains the trivial fibration between them. I was thinking that both those fibers are defined by pullbacks which are actually homotopy pullbacks and maybe we can find a map of diagram in which all component are weak equivalences to have that those two fibers are weakly equivalent. However I can't find those map and it would not show the that this map is a fibration.
simplicial-stuff higher-category-theory
simplicial-stuff higher-category-theory
asked Jan 8 at 18:13
Oscar P.Oscar P.
766
asked Jan 8 at 18:13
Oscar P.Oscar P.
766
asked Jan 8 at 18:13
Oscar P.Oscar P.
766
asked Jan 8 at 18:13
Oscar P.Oscar P.
766
asked Jan 8 at 18:13
Oscar P.Oscar P.
766
766
add a comment |
add a comment |
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