Glossary of stack theory
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This is a glossary of stack theory. For the definition of a stack, see stack.
See also: glossary of scheme theory
- fiber product
- A stack given for : an object over B is a triple (x, y, ψ), x in F(B), y in H(B), ψ an isomorphism in G(B); an arrow from (x, y, ψ) to (x', y', ψ') is a pair of morphisms such that . The resulting square with obvious projections does not commute; rather, it commutes up to natural isomorphism; i.e., it 2-commutes.
- inertia stack
- A morphism of stacks (over, say, the category of S-schemes) is a functor such that where are structures maps to the base category.
- property P
- Let P be a property of a scheme that is stable under base change (finite-type, proper, smooth, étale, etc.). Then a representable morphism is said to have propert P if, for any with B a scheme, the base change has property P.
- quotient stack
- Usually denoted by [X/G], a quotient stack generalizes a quotient of a scheme or variety.
- representable morphism
- A morphism of stacks such that, for any morphism from a scheme B, the base change is a scheme. (Some authors impose algebraic space conditions.)
(See also the reference section at stack.)
- Martin's Olsson's course notes written by Anton, http://math.berkeley.edu/~anton/written/Stacks/Stacks.pdf
- A book worked out by many authors.
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