Examples
- If the fibers of a stack are sets (meaning categories whose only morphisms are identity maps) then the stack is essentially the same as a sheaf of sets. This shows that a stack is a sort of generalization of a sheaf, taking values in arbitrary categories rather than sets.
- The stack of vector bundles on topological spaces. The condition that this is a fibered category just means that one can take pullbacks of vector bundles over continuous maps of topological spaces, and the condition that a descent datum is effective essentially means that one can construct a vector bundle over a space by gluing together vector bundles on elements of an open cover.
- The stack of quasi-coherent sheaves on schemes (with respect to the fpqc-topology and weaker topologies)
- The stack of affine schemes on a base scheme (again with respect to the fpqc topology or a weaker one)
- The moduli stack of stable curves.
Read more about this topic: Stack (descent Theory)
Famous quotes containing the word examples:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)