Stacks for Everybody
Let G be a category with a Grothendieck topology. A stack over G is a category fibered in groupoids over G, such that isomorphisms form a sheaf and every descent datum is effective. If G is the category of schemes with the étale topology, a stack is algebraic in the sense of Deligne-Mumford (respectively Artin) if it has an étale (resp. smooth) presentation.
I will try to explain the previous definitions so as to make them accessible to the widest possible audience. In order to do this, we will keep in mind one fixed example, that of vector bundles; if you know what pullback of vector bundles is in some geometric context (schemes, complex analytic spaces, but also varieties or manifolds) you should be able to follow this exposition.
KeywordsVector Bundle Open Covering Fiber Product Frame Bundle Algebraic Space
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- G. Laumon and L. Moret-BaillyChamps algébriques, Springer-Verlag, 1999.Google Scholar