How to prove $text{Sh}_G(X)simeq text{Sh}(Gbackslash X)$ when $X$ is a free $G$-space?
$begingroup$
Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.
Let $G$ be a topological group which act on $X$ continuously from the left. Consider the simplicial space $[Gbackslash X]_{cdot}$ where
$$
[Gbackslash X]_n=underbrace{Gtimes ldots times G}_{ntext{ copies of }Gtext{'s}}times X
$$
with structural maps
$$
d_0(g_1,ldots, g_n,x)=(g_2,ldots,g_n,g_1^{-1}x);
$$
$$
d_i(g_1,ldots, g_n,x)=(g_1,ldots, g_ig_{i+1},ldots, g_n,x), ~1leq ileq n-1;
$$
$$
d_n(g_1,ldots, g_n,x)=(g_1,ldots, g_{n-1},x);
$$
and
$$
s_0(g_1,ldots, g_n,x)=(e,g_1,ldots, g_n,x);
$$
$$
s_i(g_1,ldots, g_n,x)=(g_1,ldots, g_i,e , g_{i+1},ldots, g_nx),~1leq ileq n-1;
$$
$$
s_n(g_1,ldots, g_n,x)=(g_1,ldots, g_n,e,x).
$$
A $G$-equivariant sheaf on $X$ is a pair $(mathcal{F},theta)$, where $mathcal{F}in text{Sh}(X)$ and $theta$ is an isomorphism
$$
theta: d_0^*mathcal{F}overset{sim}{to} d_1^*mathcal{F},
$$
satisfying the cocycle condition
$$
d_2^*thetacirc d_0^*theta=d_1^*theta, text{ and } s_0^*theta=text{id}_{mathcal{F}}.
$$
We denote the category of $G$-equivariant sheaves on $X$ by $text{Sh}_G(X)$.
If the $G$-action on $X$ is free, then we should have a category equivalence
$$
text{Sh}(Gbackslash X)overset{sim}{to}text{Sh}_G(X).
$$
This result seems to be well-known and we do not require $G$ is compact.
My question is: how to prove this equivalence? Is there a reference with detailed proof?
reference-request representation-theory sheaf-theory group-actions
edited Jan 1 at 20:17
Shaun
9,804113684
asked Jan 1 at 19:50
Zhaoting WeiZhaoting Wei
37418
$endgroup$
add a comment |
$begingroup$
Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.
Let $G$ be a topological group which act on $X$ continuously from the left. Consider the simplicial space $[Gbackslash X]_{cdot}$ where
$$
[Gbackslash X]_n=underbrace{Gtimes ldots times G}_{ntext{ copies of }Gtext{'s}}times X
$$
with structural maps
$$
d_0(g_1,ldots, g_n,x)=(g_2,ldots,g_n,g_1^{-1}x);
$$
$$
d_i(g_1,ldots, g_n,x)=(g_1,ldots, g_ig_{i+1},ldots, g_n,x), ~1leq ileq n-1;
$$
$$
d_n(g_1,ldots, g_n,x)=(g_1,ldots, g_{n-1},x);
$$
and
$$
s_0(g_1,ldots, g_n,x)=(e,g_1,ldots, g_n,x);
$$
$$
s_i(g_1,ldots, g_n,x)=(g_1,ldots, g_i,e , g_{i+1},ldots, g_nx),~1leq ileq n-1;
$$
$$
s_n(g_1,ldots, g_n,x)=(g_1,ldots, g_n,e,x).
$$
A $G$-equivariant sheaf on $X$ is a pair $(mathcal{F},theta)$, where $mathcal{F}in text{Sh}(X)$ and $theta$ is an isomorphism
$$
theta: d_0^*mathcal{F}overset{sim}{to} d_1^*mathcal{F},
$$
satisfying the cocycle condition
$$
d_2^*thetacirc d_0^*theta=d_1^*theta, text{ and } s_0^*theta=text{id}_{mathcal{F}}.
$$
We denote the category of $G$-equivariant sheaves on $X$ by $text{Sh}_G(X)$.
If the $G$-action on $X$ is free, then we should have a category equivalence
$$
text{Sh}(Gbackslash X)overset{sim}{to}text{Sh}_G(X).
$$
This result seems to be well-known and we do not require $G$ is compact.
My question is: how to prove this equivalence? Is there a reference with detailed proof?
reference-request representation-theory sheaf-theory group-actions
edited Jan 1 at 20:17
Shaun
9,804113684
asked Jan 1 at 19:50
Zhaoting WeiZhaoting Wei
37418
$endgroup$
add a comment |
$begingroup$
Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.
Let $G$ be a topological group which act on $X$ continuously from the left. Consider the simplicial space $[Gbackslash X]_{cdot}$ where
$$
[Gbackslash X]_n=underbrace{Gtimes ldots times G}_{ntext{ copies of }Gtext{'s}}times X
$$
with structural maps
$$
d_0(g_1,ldots, g_n,x)=(g_2,ldots,g_n,g_1^{-1}x);
$$
$$
d_i(g_1,ldots, g_n,x)=(g_1,ldots, g_ig_{i+1},ldots, g_n,x), ~1leq ileq n-1;
$$
$$
d_n(g_1,ldots, g_n,x)=(g_1,ldots, g_{n-1},x);
$$
and
$$
s_0(g_1,ldots, g_n,x)=(e,g_1,ldots, g_n,x);
$$
$$
s_i(g_1,ldots, g_n,x)=(g_1,ldots, g_i,e , g_{i+1},ldots, g_nx),~1leq ileq n-1;
$$
$$
s_n(g_1,ldots, g_n,x)=(g_1,ldots, g_n,e,x).
$$
A $G$-equivariant sheaf on $X$ is a pair $(mathcal{F},theta)$, where $mathcal{F}in text{Sh}(X)$ and $theta$ is an isomorphism
$$
theta: d_0^*mathcal{F}overset{sim}{to} d_1^*mathcal{F},
$$
satisfying the cocycle condition
$$
d_2^*thetacirc d_0^*theta=d_1^*theta, text{ and } s_0^*theta=text{id}_{mathcal{F}}.
$$
We denote the category of $G$-equivariant sheaves on $X$ by $text{Sh}_G(X)$.
If the $G$-action on $X$ is free, then we should have a category equivalence
$$
text{Sh}(Gbackslash X)overset{sim}{to}text{Sh}_G(X).
$$
This result seems to be well-known and we do not require $G$ is compact.
My question is: how to prove this equivalence? Is there a reference with detailed proof?
reference-request representation-theory sheaf-theory group-actions
edited Jan 1 at 20:17
Shaun
9,804113684
asked Jan 1 at 19:50
Zhaoting WeiZhaoting Wei
37418
$endgroup$
Let $k$ be a field and $X$ be a topological space. We consider Sh$(X)$, the category of sheaves of $k$-vector spaces on $X$.
Let $G$ be a topological group which act on $X$ continuously from the left. Consider the simplicial space $[Gbackslash X]_{cdot}$ where
$$
[Gbackslash X]_n=underbrace{Gtimes ldots times G}_{ntext{ copies of }Gtext{'s}}times X
$$
with structural maps
$$
d_0(g_1,ldots, g_n,x)=(g_2,ldots,g_n,g_1^{-1}x);
$$
$$
d_i(g_1,ldots, g_n,x)=(g_1,ldots, g_ig_{i+1},ldots, g_n,x), ~1leq ileq n-1;
$$
$$
d_n(g_1,ldots, g_n,x)=(g_1,ldots, g_{n-1},x);
$$
and
$$
s_0(g_1,ldots, g_n,x)=(e,g_1,ldots, g_n,x);
$$
$$
s_i(g_1,ldots, g_n,x)=(g_1,ldots, g_i,e , g_{i+1},ldots, g_nx),~1leq ileq n-1;
$$
$$
s_n(g_1,ldots, g_n,x)=(g_1,ldots, g_n,e,x).
$$
A $G$-equivariant sheaf on $X$ is a pair $(mathcal{F},theta)$, where $mathcal{F}in text{Sh}(X)$ and $theta$ is an isomorphism
$$
theta: d_0^*mathcal{F}overset{sim}{to} d_1^*mathcal{F},
$$
satisfying the cocycle condition
$$
d_2^*thetacirc d_0^*theta=d_1^*theta, text{ and } s_0^*theta=text{id}_{mathcal{F}}.
$$
We denote the category of $G$-equivariant sheaves on $X$ by $text{Sh}_G(X)$.
If the $G$-action on $X$ is free, then we should have a category equivalence
$$
text{Sh}(Gbackslash X)overset{sim}{to}text{Sh}_G(X).
$$
This result seems to be well-known and we do not require $G$ is compact.
My question is: how to prove this equivalence? Is there a reference with detailed proof?
reference-request representation-theory sheaf-theory group-actions
reference-request representation-theory sheaf-theory group-actions
edited Jan 1 at 20:17
Shaun
9,804113684
asked Jan 1 at 19:50
Zhaoting WeiZhaoting Wei
37418
edited Jan 1 at 20:17
Shaun
9,804113684
asked Jan 1 at 19:50
Zhaoting WeiZhaoting Wei
37418
edited Jan 1 at 20:17
Shaun
9,804113684
edited Jan 1 at 20:17
Shaun
9,804113684
edited Jan 1 at 20:17
Shaun
9,804113684
9,804113684
asked Jan 1 at 19:50
Zhaoting WeiZhaoting Wei
37418
asked Jan 1 at 19:50
Zhaoting WeiZhaoting Wei
37418
asked Jan 1 at 19:50
Zhaoting WeiZhaoting Wei
37418
37418
add a comment |
add a comment |
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