Prestack

In <a href="/facts/Algebraic_geometry/rh7RHVp3">algebraic geometry</a>, a prestack F over a category C equipped with some <a href="/facts/Grothendieck_topology/Przxx5sH">Grothendieck topology</a> is a category together with a functor p: F → C satisfying a <a href="/facts/Fibered_category/LBN8vcbG">certain lifting condition</a> and such that (when the fibers are groupoids) locally isomorphic objects are isomorphic. A <a href="/facts/Stack_(mathematics)/MkASN1rA">stack</a> is a prestack with effective descents, meaning local objects may be patched together to become a global object.
Prestacks that appear in nature are typically stacks but some naively constructed prestacks (e.g., <a href="/facts/Groupoid_scheme/x3pXQJLx">groupoid scheme</a> or the prestack of <a href="/facts/Projectivized_vector_bundle/W7jjGmlW">projectivized vector bundles</a>) may not be stacks. Prestacks may be studied on their own or passed to stacks.
Since a stack is a prestack, all the results on prestacks are valid for stacks as well. Throughout the article, we work with a fixed base category C; for example, C can be the category of all schemes over some fixed scheme equipped with some <a href="/facts/Grothendieck_topology/Przxx5sH">Grothendieck topology</a>.

Prestack open-in-new

Prestack