Moduli stack of vector bundles: Difference between revisions

Revision as of 01:54, 11 June 2017

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# a morphism <math>(U, E) \to (V, F)</math> consists of <math>f: U \to V</math> in ''C'' and an isomorphism <math>f^* F \overset{\sim}\to E</math>.

Let <math>p: \operatorname{Vect}_n \to C</math> be the forgetful functor. Via ''p'', <math>\operatorname{Vect}_n</math> is a prestack over ''C''. That it is a stack over ''C'' is precisely the statement "vector bundles have the [[descent (mathematics)|descent]] property". Note that each fiber <math>\operatorname{Vect}_n(U) = p^{-1}(U)</math> over ''U'' is the category of rank-''n'' vector bundles over ''U'' where every morphism is an isomorphism (i.e., each fiber of ''p'' is a groupoid).

== See also ==