Mixing
Categories Fibered in Groupoids and Yoneda
$begingroup$
My question refers to an article of Aaron Mazel-Gee about fibered categories in grupoids where the author introduces in a quite strange way a new category ${X_0/X_1}$ providing a functor $p: {X_0/X_1} to C$ making ${X_0/X_1}$ into a category fibered in groupoids: https://etale.site/writing/stax-seminar-talk.pdf
We have following construction:
We start with a fixed category $C$ and consider the pair $(X_0,X_1)$ of two objects $X_1, X_1 in C$ satisfying identities between maps $s, t , epsilon, i $ and $m$ as described in the article. Here the excerpt:
Then comes the cruical point:
We take a $U in C$ and define a category ${X_0(U)/X_1(U)}$.
The objects are defined via $ob({X_0(U)/X_1(U)}):= X_0(U)$.
And exactly this is the problem: What is exactly $X_0(U)$? Especially how $X_0$ "acts" on $U in C$.
Some days ago I asked a similar question and got two answers from @Victoria M and @Kevin Carlson. But up to now I'm quite not sure if I understood the explanations correctly. Here the link: Fibered Categories in Groupoids
Now I would like to try to explain how I understood it and I would be glad if anybody could look thought my interpretation attempts and take corrections if there is some point which I totally misunderstood:
In the setting as above the $X_i in C$ themselves form a groupoid $(X_0,X_1)$ by definition iff they satisfy the identities between $s, t, epsilon, i ,m$ (in sense of "internal groupoid").
For arbitrary $U in C$ the only meaning for $X_i(U)$ seems logically to me comes with a "double interpretation" of the $X_i$:
as roughly elements of $C$ and as functors $X_i: C to Grp$ given formally concretely via the $Hom$ functor $C(-, X_i)$.
Seems reasonable in light of Yoneda-lemma or more precisely Yoneda embedding. Then the expresion $X_0(U)$ is identified with $C(U, X_0)$. Seems plausible.
But is this interpretation of the category $X_0/X_1$ and $X_0(U)$ exacly that what the author meant or a failable attemp to find a interpretation?
algebraic-geometry category-theory groupoids pullback
asked Jan 28 at 19:48
KarlPeterKarlPeter
5611316
$endgroup$
add a comment |
$begingroup$
My question refers to an article of Aaron Mazel-Gee about fibered categories in grupoids where the author introduces in a quite strange way a new category ${X_0/X_1}$ providing a functor $p: {X_0/X_1} to C$ making ${X_0/X_1}$ into a category fibered in groupoids: https://etale.site/writing/stax-seminar-talk.pdf
We have following construction:
We start with a fixed category $C$ and consider the pair $(X_0,X_1)$ of two objects $X_1, X_1 in C$ satisfying identities between maps $s, t , epsilon, i $ and $m$ as described in the article. Here the excerpt:
Then comes the cruical point:
We take a $U in C$ and define a category ${X_0(U)/X_1(U)}$.
The objects are defined via $ob({X_0(U)/X_1(U)}):= X_0(U)$.
And exactly this is the problem: What is exactly $X_0(U)$? Especially how $X_0$ "acts" on $U in C$.
Some days ago I asked a similar question and got two answers from @Victoria M and @Kevin Carlson. But up to now I'm quite not sure if I understood the explanations correctly. Here the link: Fibered Categories in Groupoids
Now I would like to try to explain how I understood it and I would be glad if anybody could look thought my interpretation attempts and take corrections if there is some point which I totally misunderstood:
In the setting as above the $X_i in C$ themselves form a groupoid $(X_0,X_1)$ by definition iff they satisfy the identities between $s, t, epsilon, i ,m$ (in sense of "internal groupoid").
For arbitrary $U in C$ the only meaning for $X_i(U)$ seems logically to me comes with a "double interpretation" of the $X_i$:
as roughly elements of $C$ and as functors $X_i: C to Grp$ given formally concretely via the $Hom$ functor $C(-, X_i)$.
Seems reasonable in light of Yoneda-lemma or more precisely Yoneda embedding. Then the expresion $X_0(U)$ is identified with $C(U, X_0)$. Seems plausible.
But is this interpretation of the category $X_0/X_1$ and $X_0(U)$ exacly that what the author meant or a failable attemp to find a interpretation?
algebraic-geometry category-theory groupoids pullback
asked Jan 28 at 19:48
KarlPeterKarlPeter
5611316
$endgroup$
$begingroup$
I think this is the correct interpretation, this is also the way I interpreted it.
$endgroup$
– Idéophage
Jan 31 at 1:52
$begingroup$
@Idéophage: btw: I forgot to add the link to the previous question which contains the answers I mentioned above
$endgroup$
– KarlPeter
Jan 31 at 1:57
add a comment |
$begingroup$
My question refers to an article of Aaron Mazel-Gee about fibered categories in grupoids where the author introduces in a quite strange way a new category ${X_0/X_1}$ providing a functor $p: {X_0/X_1} to C$ making ${X_0/X_1}$ into a category fibered in groupoids: https://etale.site/writing/stax-seminar-talk.pdf
We have following construction:
We start with a fixed category $C$ and consider the pair $(X_0,X_1)$ of two objects $X_1, X_1 in C$ satisfying identities between maps $s, t , epsilon, i $ and $m$ as described in the article. Here the excerpt:
Then comes the cruical point:
We take a $U in C$ and define a category ${X_0(U)/X_1(U)}$.
The objects are defined via $ob({X_0(U)/X_1(U)}):= X_0(U)$.
And exactly this is the problem: What is exactly $X_0(U)$? Especially how $X_0$ "acts" on $U in C$.
Some days ago I asked a similar question and got two answers from @Victoria M and @Kevin Carlson. But up to now I'm quite not sure if I understood the explanations correctly. Here the link: Fibered Categories in Groupoids
Now I would like to try to explain how I understood it and I would be glad if anybody could look thought my interpretation attempts and take corrections if there is some point which I totally misunderstood:
In the setting as above the $X_i in C$ themselves form a groupoid $(X_0,X_1)$ by definition iff they satisfy the identities between $s, t, epsilon, i ,m$ (in sense of "internal groupoid").
For arbitrary $U in C$ the only meaning for $X_i(U)$ seems logically to me comes with a "double interpretation" of the $X_i$:
as roughly elements of $C$ and as functors $X_i: C to Grp$ given formally concretely via the $Hom$ functor $C(-, X_i)$.
Seems reasonable in light of Yoneda-lemma or more precisely Yoneda embedding. Then the expresion $X_0(U)$ is identified with $C(U, X_0)$. Seems plausible.
But is this interpretation of the category $X_0/X_1$ and $X_0(U)$ exacly that what the author meant or a failable attemp to find a interpretation?
algebraic-geometry category-theory groupoids pullback
asked Jan 28 at 19:48
KarlPeterKarlPeter
5611316
$endgroup$
My question refers to an article of Aaron Mazel-Gee about fibered categories in grupoids where the author introduces in a quite strange way a new category ${X_0/X_1}$ providing a functor $p: {X_0/X_1} to C$ making ${X_0/X_1}$ into a category fibered in groupoids: https://etale.site/writing/stax-seminar-talk.pdf
We have following construction:
We start with a fixed category $C$ and consider the pair $(X_0,X_1)$ of two objects $X_1, X_1 in C$ satisfying identities between maps $s, t , epsilon, i $ and $m$ as described in the article. Here the excerpt:
Then comes the cruical point:
We take a $U in C$ and define a category ${X_0(U)/X_1(U)}$.
The objects are defined via $ob({X_0(U)/X_1(U)}):= X_0(U)$.
And exactly this is the problem: What is exactly $X_0(U)$? Especially how $X_0$ "acts" on $U in C$.
Some days ago I asked a similar question and got two answers from @Victoria M and @Kevin Carlson. But up to now I'm quite not sure if I understood the explanations correctly. Here the link: Fibered Categories in Groupoids
Now I would like to try to explain how I understood it and I would be glad if anybody could look thought my interpretation attempts and take corrections if there is some point which I totally misunderstood:
In the setting as above the $X_i in C$ themselves form a groupoid $(X_0,X_1)$ by definition iff they satisfy the identities between $s, t, epsilon, i ,m$ (in sense of "internal groupoid").
For arbitrary $U in C$ the only meaning for $X_i(U)$ seems logically to me comes with a "double interpretation" of the $X_i$:
as roughly elements of $C$ and as functors $X_i: C to Grp$ given formally concretely via the $Hom$ functor $C(-, X_i)$.
Seems reasonable in light of Yoneda-lemma or more precisely Yoneda embedding. Then the expresion $X_0(U)$ is identified with $C(U, X_0)$. Seems plausible.
But is this interpretation of the category $X_0/X_1$ and $X_0(U)$ exacly that what the author meant or a failable attemp to find a interpretation?
algebraic-geometry category-theory groupoids pullback
algebraic-geometry category-theory groupoids pullback
asked Jan 28 at 19:48
KarlPeterKarlPeter
5611316
asked Jan 28 at 19:48
KarlPeterKarlPeter
5611316
edited Jan 31 at 1:55
KarlPeter
asked Jan 28 at 19:48
KarlPeterKarlPeter
5611316
asked Jan 28 at 19:48
KarlPeterKarlPeter
5611316
asked Jan 28 at 19:48
KarlPeterKarlPeter
5611316
5611316
$begingroup$
I think this is the correct interpretation, this is also the way I interpreted it.
$endgroup$
– Idéophage
Jan 31 at 1:52
$begingroup$
@Idéophage: btw: I forgot to add the link to the previous question which contains the answers I mentioned above
$endgroup$
– KarlPeter
Jan 31 at 1:57
add a comment |
$begingroup$
I think this is the correct interpretation, this is also the way I interpreted it.
$endgroup$
– Idéophage
Jan 31 at 1:52
$begingroup$
@Idéophage: btw: I forgot to add the link to the previous question which contains the answers I mentioned above
$endgroup$
– KarlPeter
Jan 31 at 1:57
$begingroup$
I think this is the correct interpretation, this is also the way I interpreted it.
$endgroup$
– Idéophage
Jan 31 at 1:52
$begingroup$
I think this is the correct interpretation, this is also the way I interpreted it.
$endgroup$
– Idéophage
Jan 31 at 1:52
$begingroup$
@Idéophage: btw: I forgot to add the link to the previous question which contains the answers I mentioned above
$endgroup$
– KarlPeter
Jan 31 at 1:57
$begingroup$
@Idéophage: btw: I forgot to add the link to the previous question which contains the answers I mentioned above
$endgroup$
– KarlPeter
Jan 31 at 1:57
add a comment |
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$begingroup$
I think this is the correct interpretation, this is also the way I interpreted it.
$endgroup$
– Idéophage
Jan 31 at 1:52
$begingroup$
@Idéophage: btw: I forgot to add the link to the previous question which contains the answers I mentioned above
$endgroup$
– KarlPeter
Jan 31 at 1:57