Etale groupoid and Morita equivalence
$begingroup$
Let $mathcal{G}=(G_{1}rightrightarrows G_0)$ be a groupoid, where $G_{0}$ is the space of objects and $G_{1}$ is the space of morphisms. $mathcal{G}$ is called etale if both the source and target maps
$$
s,t:G_{1}rightarrow G_{0}
$$
are local diffeomorphisms. Being etale is $not$ invariant under Morita equivalence (equivalence of categories).
Could anyone give me a simple example of this fact?
category-theory groupoids orbifolds
edited Jan 13 at 6:47
Praphulla Koushik
203119
asked Aug 18 '12 at 2:33
T. PetroT. Petro
161
$endgroup$
add a comment |
$begingroup$
Let $mathcal{G}=(G_{1}rightrightarrows G_0)$ be a groupoid, where $G_{0}$ is the space of objects and $G_{1}$ is the space of morphisms. $mathcal{G}$ is called etale if both the source and target maps
$$
s,t:G_{1}rightarrow G_{0}
$$
are local diffeomorphisms. Being etale is $not$ invariant under Morita equivalence (equivalence of categories).
Could anyone give me a simple example of this fact?
category-theory groupoids orbifolds
edited Jan 13 at 6:47
Praphulla Koushik
203119
asked Aug 18 '12 at 2:33
T. PetroT. Petro
161
$endgroup$
add a comment |
$begingroup$
Let $mathcal{G}=(G_{1}rightrightarrows G_0)$ be a groupoid, where $G_{0}$ is the space of objects and $G_{1}$ is the space of morphisms. $mathcal{G}$ is called etale if both the source and target maps
$$
s,t:G_{1}rightarrow G_{0}
$$
are local diffeomorphisms. Being etale is $not$ invariant under Morita equivalence (equivalence of categories).
Could anyone give me a simple example of this fact?
category-theory groupoids orbifolds
edited Jan 13 at 6:47
Praphulla Koushik
203119
asked Aug 18 '12 at 2:33
T. PetroT. Petro
161
$endgroup$
Let $mathcal{G}=(G_{1}rightrightarrows G_0)$ be a groupoid, where $G_{0}$ is the space of objects and $G_{1}$ is the space of morphisms. $mathcal{G}$ is called etale if both the source and target maps
$$
s,t:G_{1}rightarrow G_{0}
$$
are local diffeomorphisms. Being etale is $not$ invariant under Morita equivalence (equivalence of categories).
Could anyone give me a simple example of this fact?
category-theory groupoids orbifolds
category-theory groupoids orbifolds
edited Jan 13 at 6:47
Praphulla Koushik
203119
asked Aug 18 '12 at 2:33
T. PetroT. Petro
161
edited Jan 13 at 6:47
Praphulla Koushik
203119
asked Aug 18 '12 at 2:33
T. PetroT. Petro
161
edited Jan 13 at 6:47
Praphulla Koushik
203119
edited Jan 13 at 6:47
Praphulla Koushik
203119
edited Jan 13 at 6:47
Praphulla Koushik
203119
203119
asked Aug 18 '12 at 2:33
T. PetroT. Petro
161
asked Aug 18 '12 at 2:33
T. PetroT. Petro
161
asked Aug 18 '12 at 2:33
T. PetroT. Petro
161
161
add a comment |
add a comment |
1 Answer
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$begingroup$
Take the action of $R$ on itself. The associated groupoid is the equivalence relation.
This is Morita equivalent to a point (the action is free and proper). $Rtimes R$ is not etale.
A better example would be to look at the full holonomy groupoid vs the restriction to a complete transversal.
answered Aug 23 '13 at 10:53
user91475user91475
212
$endgroup$
add a comment |
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1 Answer
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1 Answer
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votes
$begingroup$
Take the action of $R$ on itself. The associated groupoid is the equivalence relation.
This is Morita equivalent to a point (the action is free and proper). $Rtimes R$ is not etale.
A better example would be to look at the full holonomy groupoid vs the restriction to a complete transversal.
answered Aug 23 '13 at 10:53
user91475user91475
212
$endgroup$
add a comment |
$begingroup$
Take the action of $R$ on itself. The associated groupoid is the equivalence relation.
This is Morita equivalent to a point (the action is free and proper). $Rtimes R$ is not etale.
A better example would be to look at the full holonomy groupoid vs the restriction to a complete transversal.
answered Aug 23 '13 at 10:53
user91475user91475
212
$endgroup$
add a comment |
$begingroup$
Take the action of $R$ on itself. The associated groupoid is the equivalence relation.
This is Morita equivalent to a point (the action is free and proper). $Rtimes R$ is not etale.
A better example would be to look at the full holonomy groupoid vs the restriction to a complete transversal.
answered Aug 23 '13 at 10:53
user91475user91475
212
$endgroup$
Take the action of $R$ on itself. The associated groupoid is the equivalence relation.
This is Morita equivalent to a point (the action is free and proper). $Rtimes R$ is not etale.
A better example would be to look at the full holonomy groupoid vs the restriction to a complete transversal.
answered Aug 23 '13 at 10:53
user91475user91475
212
answered Aug 23 '13 at 10:53
user91475user91475
212
answered Aug 23 '13 at 10:53
user91475user91475
212
answered Aug 23 '13 at 10:53
user91475user91475
212
212
add a comment |
add a comment |
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