Lesson plan
Prestack
In algebraic geometry, a prestack F over a category C equipped with some Grothendieck topology is a category together with a functor p: F → C satisfying a certain lifting condition and such that (when the fibers are groupoids) locally isomorphic objects are isomorphic. A stack 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., groupoid scheme or the prestack of projectivized vector bundles) 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 Grothendieck topology.
Article summary
In algebraic geometry, a prestack F over a category C equipped with some Grothendieck topology is a category together with a functor p: F → C satisfying a certain lifting condition and such that (when the fibers are groupoids) locally isomorphic objects are isomorphic. A stack 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., groupoid scheme or the prestack of projectivized vector bundles) 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 Grothendieck topology.
Key claims
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Discussion questions
- Summarise the article in your own words. Which single claim would you verify first, and why?
- Which claim in this article is most likely to be revised as the sources update? How would you check its current status?
- If two sources disagreed about one of these facts, what would you need in order to trust one over the other?
Bibliography
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Key claims and bibliography are generated from the same tracked claims and sources as the article page.