Descent data and trivialization of bundles via coherent isomorphisms of fibers
$begingroup$
In this MO question I tried to understand how a trivialization of a bundle (continuous map) $begin{smallmatrix}A\ downarrow\ B end{smallmatrix}$ is related to a coherent family of isomorphisms between its fibers.
I have a tentative answer. My problem is that I can't quite show that descent data for $begin{smallmatrix}A\ downarrow\ B end{smallmatrix}$ along $Bto bf 1$ is the same as a coherent family of isomorphisms between the fibers.
Already the transition isomorphism should be a bundle isomorphism $alphatimes 1_Bcong 1_Btimes alpha$ viewed as bundles $begin{smallmatrix}Atimes B\ downarrow\ Btimes B end{smallmatrix},begin{smallmatrix}Btimes A\ downarrow\ Btimes B end{smallmatrix}$. Evidently the respective fibers at $(b,b^prime)$ are $alpha^{-1}(b)times B,Btimes alpha^{-1}(b^prime)$, so I guess some commutativity conditions will ensure the isomorphism descends to isomorphisms $alpha^{-1}(b)cong alpha^{-1}(b^prime)$. However, I don't see how this happens formally, and I'm not even at the stage of translating the cocycle condition into coherence of the fiber isomorphisms.
I would like some help in working out the details here. The relevant reference (without details) is in the first link.
general-topology category-theory fiber-bundles descent
asked Dec 16 '18 at 22:49
ArrowArrow
5,19931446
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In this MO question I tried to understand how a trivialization of a bundle (continuous map) $begin{smallmatrix}A\ downarrow\ B end{smallmatrix}$ is related to a coherent family of isomorphisms between its fibers.
I have a tentative answer. My problem is that I can't quite show that descent data for $begin{smallmatrix}A\ downarrow\ B end{smallmatrix}$ along $Bto bf 1$ is the same as a coherent family of isomorphisms between the fibers.
Already the transition isomorphism should be a bundle isomorphism $alphatimes 1_Bcong 1_Btimes alpha$ viewed as bundles $begin{smallmatrix}Atimes B\ downarrow\ Btimes B end{smallmatrix},begin{smallmatrix}Btimes A\ downarrow\ Btimes B end{smallmatrix}$. Evidently the respective fibers at $(b,b^prime)$ are $alpha^{-1}(b)times B,Btimes alpha^{-1}(b^prime)$, so I guess some commutativity conditions will ensure the isomorphism descends to isomorphisms $alpha^{-1}(b)cong alpha^{-1}(b^prime)$. However, I don't see how this happens formally, and I'm not even at the stage of translating the cocycle condition into coherence of the fiber isomorphisms.
I would like some help in working out the details here. The relevant reference (without details) is in the first link.
general-topology category-theory fiber-bundles descent
asked Dec 16 '18 at 22:49
ArrowArrow
5,19931446
$endgroup$
add a comment |
$begingroup$
In this MO question I tried to understand how a trivialization of a bundle (continuous map) $begin{smallmatrix}A\ downarrow\ B end{smallmatrix}$ is related to a coherent family of isomorphisms between its fibers.
I have a tentative answer. My problem is that I can't quite show that descent data for $begin{smallmatrix}A\ downarrow\ B end{smallmatrix}$ along $Bto bf 1$ is the same as a coherent family of isomorphisms between the fibers.
Already the transition isomorphism should be a bundle isomorphism $alphatimes 1_Bcong 1_Btimes alpha$ viewed as bundles $begin{smallmatrix}Atimes B\ downarrow\ Btimes B end{smallmatrix},begin{smallmatrix}Btimes A\ downarrow\ Btimes B end{smallmatrix}$. Evidently the respective fibers at $(b,b^prime)$ are $alpha^{-1}(b)times B,Btimes alpha^{-1}(b^prime)$, so I guess some commutativity conditions will ensure the isomorphism descends to isomorphisms $alpha^{-1}(b)cong alpha^{-1}(b^prime)$. However, I don't see how this happens formally, and I'm not even at the stage of translating the cocycle condition into coherence of the fiber isomorphisms.
I would like some help in working out the details here. The relevant reference (without details) is in the first link.
general-topology category-theory fiber-bundles descent
asked Dec 16 '18 at 22:49
ArrowArrow
5,19931446
$endgroup$
In this MO question I tried to understand how a trivialization of a bundle (continuous map) $begin{smallmatrix}A\ downarrow\ B end{smallmatrix}$ is related to a coherent family of isomorphisms between its fibers.
I have a tentative answer. My problem is that I can't quite show that descent data for $begin{smallmatrix}A\ downarrow\ B end{smallmatrix}$ along $Bto bf 1$ is the same as a coherent family of isomorphisms between the fibers.
Already the transition isomorphism should be a bundle isomorphism $alphatimes 1_Bcong 1_Btimes alpha$ viewed as bundles $begin{smallmatrix}Atimes B\ downarrow\ Btimes B end{smallmatrix},begin{smallmatrix}Btimes A\ downarrow\ Btimes B end{smallmatrix}$. Evidently the respective fibers at $(b,b^prime)$ are $alpha^{-1}(b)times B,Btimes alpha^{-1}(b^prime)$, so I guess some commutativity conditions will ensure the isomorphism descends to isomorphisms $alpha^{-1}(b)cong alpha^{-1}(b^prime)$. However, I don't see how this happens formally, and I'm not even at the stage of translating the cocycle condition into coherence of the fiber isomorphisms.
I would like some help in working out the details here. The relevant reference (without details) is in the first link.
general-topology category-theory fiber-bundles descent
general-topology category-theory fiber-bundles descent
asked Dec 16 '18 at 22:49
ArrowArrow
5,19931446
asked Dec 16 '18 at 22:49
ArrowArrow
5,19931446
asked Dec 16 '18 at 22:49
ArrowArrow
5,19931446
asked Dec 16 '18 at 22:49
ArrowArrow
5,19931446
asked Dec 16 '18 at 22:49
ArrowArrow
5,19931446
5,19931446
add a comment |
add a comment |
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