Glossary of stack theory

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This is a glossary of stack theory. For the definition of a stack, see stack.


fiber product
A stack F \times_G H given for f: F \to G, g:H \to G: an object over B is a triple (x, y, ψ), x in F(B), y in H(B), ψ an isomorphism f(x) \to g(y) in G(B); an arrow from (x, y, ψ) to (x', y', ψ') is a pair of morphisms \alpha: x \to x', \beta: y \to y' such that \psi' \circ f(\alpha) = g(\beta) \circ \psi. 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 f: F \to G of stacks (over, say, the category of S-schemes) is a functor such that f \circ P_G = P_F where P_F, P_G 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 f: F \to G is said to have propert P if, for any B \to G with B a scheme, the base change F \times_G B \to B 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 F \to G of stacks such that, for any morphism B \to G from a scheme B, the base change F \times_G B is a scheme. (Some authors impose algebraic space conditions.)


(See also the reference section at stack.)