Stack (category theory)

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In algebraic geometry, an algebraic stack is a concept introduced to generalize algebraic varieties, schemes, and algebraic spaces. They were originally proposed in a paper of Deligne and Mumford on the compactifications of moduli of curves; such stacks are now called Deligne-Mumford stacks. They were later generalized by Michael Artin to what is now called an Artin stack, or sometimes, confusingly, an algebraic stack.

There are two main classes of stacks, Deligne-Mumford stacks (DM stacks) and Artin stacks. The initial construction requires a category fibered in groupoids (CFG), and adds two conditions

a requirement for extending diagrams via cartesian squares, and
the ability to "glue" morphisms and objects, similar to the gluing of local affine spaces into schemes.

More generally a stack refers to any category acting more or less like a moduli space with a universal family (analogous to a classifying space) parametrizing a family of related mathematical objects such as schemes or topological spaces, especially when the members of these families have nontrivial automorphisms. This leads to the notion that the points of the stack should carry automorphisms themselves, and this in turn gives rise to the notion of a stack as a certain kind of "category fibered in groupoids".

Moduli spaces which do not carry this extra information are then referred to as coarse moduli spaces and stacks then act as relatively fine moduli spaces.

Examples

The moduli space of algebraic curves (Deligne-Mumford stack) defined as a universal family of curves of given genus g does not exist as an algebraic variety because in particular there are elliptic curves admitting nontrivial automorphisms. For elliptic curves over the complex numbers the corresponding stack is a geometrical factor of the upper half-plane by the action of the modular group.

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