Mixing
A certain colimit of representables sheaves, namely group actions, is a sheaf: why?
$begingroup$
In the first answer to this post on MO, one finds that
When you look at the category of sheaves on the category of finite
action with the natural topology (covering are surjection of finite
action) you gets rather trivially the topos of all continuous action
of the galois group (basically because they are justs non-finite
coproducts of the finite action).
(please read the context in order to understand my question). In brief (filling the holes in those three lines), orbits under an action of the Galois group must be finite, therefore every $G$-set is a colimit of its finite orbits. Therefore every $G$-set is a colimit of finite $G$-sets, that are elements of the site, i.e. representable. Now my question is: why are we sure that the resulting functor is a sheaf? As a colimit of representable sheaves it is a presheaf, but why a sheaf? The topology is the natural one: jointly surjective families. I have tried to convince myself that the colimit respects the glueing property, but I do not think this is the way...
Thank you in advance.
group-actions topos-theory profinite-groups
asked Jan 2 at 19:37
W. RetherW. Rether
728417
$endgroup$
add a comment |
$begingroup$
In the first answer to this post on MO, one finds that
When you look at the category of sheaves on the category of finite
action with the natural topology (covering are surjection of finite
action) you gets rather trivially the topos of all continuous action
of the galois group (basically because they are justs non-finite
coproducts of the finite action).
(please read the context in order to understand my question). In brief (filling the holes in those three lines), orbits under an action of the Galois group must be finite, therefore every $G$-set is a colimit of its finite orbits. Therefore every $G$-set is a colimit of finite $G$-sets, that are elements of the site, i.e. representable. Now my question is: why are we sure that the resulting functor is a sheaf? As a colimit of representable sheaves it is a presheaf, but why a sheaf? The topology is the natural one: jointly surjective families. I have tried to convince myself that the colimit respects the glueing property, but I do not think this is the way...
Thank you in advance.
group-actions topos-theory profinite-groups
asked Jan 2 at 19:37
W. RetherW. Rether
728417
$endgroup$
add a comment |
$begingroup$
In the first answer to this post on MO, one finds that
When you look at the category of sheaves on the category of finite
action with the natural topology (covering are surjection of finite
action) you gets rather trivially the topos of all continuous action
of the galois group (basically because they are justs non-finite
coproducts of the finite action).
(please read the context in order to understand my question). In brief (filling the holes in those three lines), orbits under an action of the Galois group must be finite, therefore every $G$-set is a colimit of its finite orbits. Therefore every $G$-set is a colimit of finite $G$-sets, that are elements of the site, i.e. representable. Now my question is: why are we sure that the resulting functor is a sheaf? As a colimit of representable sheaves it is a presheaf, but why a sheaf? The topology is the natural one: jointly surjective families. I have tried to convince myself that the colimit respects the glueing property, but I do not think this is the way...
Thank you in advance.
group-actions topos-theory profinite-groups
asked Jan 2 at 19:37
W. RetherW. Rether
728417
$endgroup$
In the first answer to this post on MO, one finds that
When you look at the category of sheaves on the category of finite
action with the natural topology (covering are surjection of finite
action) you gets rather trivially the topos of all continuous action
of the galois group (basically because they are justs non-finite
coproducts of the finite action).
(please read the context in order to understand my question). In brief (filling the holes in those three lines), orbits under an action of the Galois group must be finite, therefore every $G$-set is a colimit of its finite orbits. Therefore every $G$-set is a colimit of finite $G$-sets, that are elements of the site, i.e. representable. Now my question is: why are we sure that the resulting functor is a sheaf? As a colimit of representable sheaves it is a presheaf, but why a sheaf? The topology is the natural one: jointly surjective families. I have tried to convince myself that the colimit respects the glueing property, but I do not think this is the way...
Thank you in advance.
group-actions topos-theory profinite-groups
group-actions topos-theory profinite-groups
asked Jan 2 at 19:37
W. RetherW. Rether
728417
asked Jan 2 at 19:37
W. RetherW. Rether
728417
asked Jan 2 at 19:37
W. RetherW. Rether
728417
asked Jan 2 at 19:37
W. RetherW. Rether
728417
asked Jan 2 at 19:37
W. RetherW. Rether
728417
728417
add a comment |
add a comment |
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