Algebraic Geometry Stack


In the algebraic geometry of Grothendieck, a stack refers to a sheaf of categories. In particular, a stack is a presheaf of categories in which the following descent properties (Brylinski 1993) are satisfied:

1. Given topological spaces X and Y with a function f:Y->X and two sheaves A and B of groups on Y, the assignment (g:Z->Y)->Hom_Z(g^(-1)A,g^(-1)B) defines a sheaf on Y called Hom__(A,B);

2. Given an open subset V subset X of X, a local surjective homeomorphism f:Y->V, and a sheaf A over Y together with an isomorphism phi:p_1^(-1)A->p_2^(-1)A of sheaves over Y×Y for which the left diagram above, commutes then there exists an sheaf A^' over V (unique up to isomorphism) together with an isomorphism psi:f^(-1)A^'->A of sheaves in Y such that above right diagram of sheaf isomorphisms of Y×Y commutes.

Here, p_i:Y×Y×Y->Y denotes projection onto one of the factors and p_(ij):Y×Y×Y->Y×Y denotes projection onto two of the three factors.

See also

Category, Category Theory, Commutative Diagram, Functor, Homeomorphism, Isomorphism, List, Pop, Presheaf, Presheaf of Categories, Push, Queue, Reverse Polish Notation, Sheaf, Stack, Topological Space

This entry contributed by Christopher Stover

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Brylinski, J. Loop Spaces, Characteristic Classes and Geometric Quantization. Boston: Birkhäuser, 1993.

Cite this as:

Stover, Christopher. "Algebraic Geometry Stack." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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