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)
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