When is a sheaf of groups (algebras, rings, modules) a group (algebra, ring, module)? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T13:47:03Z http://mathoverflow.net/feeds/question/120822 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/120822/when-is-a-sheaf-of-groups-algebras-rings-modules-a-group-algebra-ring-modu When is a sheaf of groups (algebras, rings, modules) a group (algebra, ring, module)? Oscar Guajardo 2013-02-05T01:53:47Z 2013-02-05T02:16:09Z <p>If $\pi:E\to M$ is a vector bundle then the set of sections $\Gamma(E)$ is naturally a vector space under fibrewise addition and scalar multiplication on the bundle $E$. This holds similarily for bundles of algebras or modules, thought I'm not sure if it holds for bundles of groups (certainly not for principal bundles). The main example I have in mind is the algebra $\mathcal{C}^\infty (M)$ of smooth real-valued functions on a smooth manifold (just considering the ring structure of the reals, not the field structure).</p> <p>Now, given a sheaf $\mathcal{O}$ with values in some category $\mathcal{C}$, when is $\mathcal{O}_X$ an object in $\mathcal{C}$? </p> <p>As the examples above show (basically abelian groups with extra structures) this is true when $\mathcal{O}_X$ is the sheaf of sections of a fibre bundle whose fibres are objects of $\mathcal{C}$. Another relevant question would be if this is indeed the case for locally trivial fibrations only. Concretely:</p> <p>Is it true that the sheaf $\mathcal{O}_X$ is an object of $\mathcal{C}$ only when $\mathcal{O}_X$ is the sheaf of sections of a fibre bundle $B$ over $X$ whose fibres $B_x$ are objects of $\mathcal{C}$?</p> http://mathoverflow.net/questions/120822/when-is-a-sheaf-of-groups-algebras-rings-modules-a-group-algebra-ring-modu/120824#120824 Answer by Steven Landsburg for When is a sheaf of groups (algebras, rings, modules) a group (algebra, ring, module)? Steven Landsburg 2013-02-05T02:16:09Z 2013-02-05T02:16:09Z <p>A sheaf ${\cal O}$ is <i>a fortiori</i> a presheaf. This means it's a functor that takes the category of open sets in $X$ to your fixed category ${\cal C}$. So, by definition, $\Gamma(X,{\cal O})={\cal O}(X)$ is an object of ${\cal C}$.</p>