Return to Answer

$H^1(X,\mathcal G)$ where $X$ is any topological space, and $\mathcal G$ any sheaf of groups, classifies $\mathcal G$-torsors on $X$. A $G$-torsor is a sheaf of sets with a $\mathcal G$-action on $X$ which is isomorphic to the sheaf $G$ on small enough open sets.

If $\mathcal G$ is the set of locally constant functions valued is a group $G$, then a $\mathcal G$-torsor is really just an $G$-local system (this is essentially by definition), so for any group this result holds.

EDIT: Paul's answer made me realize I hadn't proceeded to then point out that a local system is the same thing as a map $\pi_1(X)\to G$ up to conjugacy, by taking the monodromy along paths (which requires choosing base points, hence the conjugacy). I wasn't trying to be mysterious; I just have this so ingrained in my head that I forgot it was worth mentioning.

You have to careful here, because when $G$ is a topological group, it's very tempting to think about the sheaf of continuous functions to $G$ on $X$; that's the sheaf of groups whose torsors are principal bundles. This will only agree with the above if $G$ is discrete (hence Angelo's answer).

$H^1(X,\mathcal G)$ where $X$ is any topological space, and $\mathcal G$ any sheaf of groups, classifies $\mathcal G$-torsors on $X$. A $\mathcal G$-torsor is a sheaf of sets with a $\mathcal G$-action on $X$ which is isomorphic to the sheaf $\mathcal G$ on small enough open sets.

If $\mathcal G$ is the set of locally constant functions valued is a group $G$, then a $\mathcal G$-torsor is really just an $G$-local system (this is essentially by definition), so for any group this result holds.

EDIT: Paul's answer made me realize I hadn't proceeded to then point out that a local system is the same thing as a map $\pi_1(X)\to G$ up to conjugacy, by taking the monodromy along paths (which requires choosing base points, hence the conjugacy). I wasn't trying to be mysterious; I just have this so ingrained in my head that I forgot it was worth mentioning.

You have to careful here, because when $G$ is a topological group, it's very tempting to think about the sheaf of continuous functions to $G$ on $X$; that's the sheaf of groups whose torsors are principal bundles. This will only agree with the above if $G$ is discrete (hence Angelo's answer).

$H^1(X,\mathcal G)$ where $X$ is any topological space, and $\mathcal G$ any sheaf of groups, classifies $\mathcal G$-torsors on $X$. A $G$-torsor is a sheaf of sets with a $\mathcal G$-action on $X$ which is isomorphic to the sheaf $G$ on small enough open sets.

If $\mathcal G$ is the set of locally constant functions valued is a group $G$, then a $\mathcal G$-torsor is really just an $G$-local system (this is essentially by definition), so for any group this result holds.

You have to careful here, because when $G$ is a topological group, it's very tempting to think about the sheaf of continuous functions to $G$ on $X$; that's the sheaf of groups whose torsors are principal bundles. This will only agree with the above if $G$ is discrete (hence Angelo's answer).

$H^1(X,\mathcal G)$ where $X$ is any topological space, and $\mathcal G$ any sheaf of groups, classifies $\mathcal G$-torsors on $X$. A $G$-torsor is a sheaf of sets with a $\mathcal G$-action on $X$ which is isomorphic to the sheaf $G$ on small enough open sets.

If $\mathcal G$ is the set of locally constant functions valued is a group $G$, then a $\mathcal G$-torsor is really just an $G$-local system (this is essentially by definition), so for any group this result holds.

EDIT: Paul's answer made me realize I hadn't proceeded to then point out that a local system is the same thing as a map $\pi_1(X)\to G$ up to conjugacy, by taking the monodromy along paths (which requires choosing base points, hence the conjugacy). I wasn't trying to be mysterious; I just have this so ingrained in my head that I forgot it was worth mentioning.

You have to careful here, because when $G$ is a topological group, it's very tempting to think about the sheaf of continuous functions to $G$ on $X$; that's the sheaf of groups whose torsors are principal bundles. This will only agree with the above if $G$ is discrete (hence Angelo's answer).

$H^1(X,\mathcal G)$ where $X$ is any topological space, and $\mathcal G$ any sheaf of groups, classifies $\mathcal G$-torsors on $X$. A $G$-torsor is a sheaf of sets with a $\mathcal G$-action on $X$ which is isomorphic to the sheaf $G$ on small enough open sets.

If $\mathcal G$ is the set of locally constant functions valued is a group $G$, then a $\mathcal G$-torsor is really just an $G$-local system (this is essentially by definition), so for any group this result holds.

You have to careful here, because it's very tempting to think about the sheaf of continuous functions to $G$ on $X$; that's the sheaf of groups whose torsors are principal bundles.

$H^1(X,\mathcal G)$ where $X$ is any topological space, and $\mathcal G$ any sheaf of groups, classifies $\mathcal G$-torsors on $X$. A $G$-torsor is a sheaf of sets with a $\mathcal G$-action on $X$ which is isomorphic to the sheaf $G$ on small enough open sets.

If $\mathcal G$ is the set of locally constant functions valued is a group $G$, then a $\mathcal G$-torsor is really just an $G$-local system (this is essentially by definition), so for any group this result holds.

You have to careful here, because when $G$ is a topological group, it's very tempting to think about the sheaf of continuous functions to $G$ on $X$; that's the sheaf of groups whose torsors are principal bundles. This will only agree with the above if $G$ is discrete (hence Angelo's answer).