stacks (mathematics)

stacks (mathematics)
Name	Stacks (mathematics)
Field	Algebraic geometry, Category theory
Introduced	1960s–1990s
Main contributors	Grothendieck, Deligne, Mumford, Artin, Giraud, Laumon, Moret-Bailly

stacks (mathematics).

Stacks are a categorical and sheaf-theoretic refinement of fibered categories designed to encode objects together with their automorphisms and descent data; they generalize sheaves, moduli functors, and gerbes to treat families with nontrivial isotropy. Originating in the work of Alexander Grothendieck, formalized by Jean Giraud, and extensively developed by Pierre Deligne, David Mumford, and Michael Artin, stacks are central in modern Alexeev, Laumon, and Moret-Bailly-style moduli problems, bridging category theory and algebraic geometry.

Introduction

Stacks arise when classical moduli schemes fail to exist because of automorphisms, as in the moduli of curves studied by David Mumford and Deligne; they simultaneously record objects and their symmetries. A stack is a fibered category over a site satisfying effective descent for objects and morphisms; examples include the moduli stack of curves, classifying stacks such as B G for a group G, and gerbes classified by cohomology classes as in Jean Giraud's work. Foundational developments tied to deformation theory and representability theorems were advanced by Michael Artin and have been applied by researchers including Pierre Deligne, David Mumford, Laumon, Moret-Bailly, Jacob Lurie, and Max Lieblich.

Definitions and examples

Formally, a stack over a site S is a category fibered in groupoids F → S satisfying descent: isomorphisms form sheaves and objects glue effectively. Key examples: the moduli stack M_g of stable curves addressed by David Mumford and Pierre Deligne; the classifying stack B G for an algebraic group G used in work by Jean-Pierre Serre and Armand Borel; the quotient stack [X/G] appearing in geometric invariant theory of David Mumford and in constructions by Michael Artin; and gerbes arising in Jean Giraud's classification by H^2. Other notable instances include the Picard stack studied by Grothendieck and the Hilbert stack in investigations by Alexander Grothendieck and Michael Artin.

Morphisms, 2-fibered products, and gerbes

Morphisms of stacks are pseudonatural transformations of fibered categories studied in bicategorical contexts developed by Jean Bénabou and John Baez. Two-stacks and 2-fibered products are constructed using 2-categorical limits; the 2-fibered product realizes base change in moduli problems as in the work of Alexander Grothendieck on descent. Gerbes are stacks locally nonempty and locally connected, classified by cohomology classes in H^2 as in Jean Giraud; classical instances include the Brauer group treated by Richard Brauer and Claude Chevalley in algebraic contexts, and torsors under group stacks as studied by Michel Demazure and Grothendieck.

Algebraic stacks and Artin/Deligne–Mumford stacks

Algebraic stacks combine stack-theoretic descent with representability conditions: diagonal representability and smooth (or étale) presentations by schemes give Artin stacks and Deligne–Mumford stacks respectively. Michael Artin formulated criteria for algebraicity, while Pierre Deligne and David Mumford introduced Deligne–Mumford stacks to treat moduli of curves with finite stabilizers. Important examples include the moduli stack M_g of stable curves (Deligne–Mumford), quotient stacks [X/G] arising in David Mumford's geometric invariant theory, and stacks of principal bundles studied by Atiyah and Bott. Deep representability results and structure theorems involve input from Alexander Grothendieck's functorial viewpoint, Michael Artin's approximation theorems, and later refinements by Laumon and Moret-Bailly.

Cohomological and deformation-theoretic aspects

Cohomology of stacks generalizes sheaf cohomology and gives invariants controlling obstructions and deformation spaces; foundational contributions came from Grothendieck and Jean-Pierre Serre. Deformation theory for stacks uses cotangent complexes and obstruction theories developed by Pierre Deligne, Illusie, and Michael Artin; virtual fundamental classes for stacks were introduced in enumerative geometry applications by Katić and extended by Behrend and Fantechi. The study of the Brauer group of stacks, gerbes, and twisted sheaves connects to work of Max Lieblich, Căldăraru, and Toën, while l-adic and étale cohomology of stacks was advanced by contributions from Pierre Deligne and Beilinson.

Applications in moduli theory and geometry

Stacks underpin modern moduli theory: compactified moduli spaces, Gromov–Witten theory, Donaldson–Thomas theory, and derived enhancements frequently use stack structures. The construction of moduli of vector bundles on curves builds on work by Atiyah, Narasimhan, and Ramanan; moduli of sheaves and complexes employ methods from Mukai and Simpson. Stacks appear in intersection theory on moduli spaces addressed by Fulton, in mirror symmetry studied by Kontsevich and Givental, and in arithmetic geometry via moduli of abelian varieties in the work of Igusa and Mumford. Derived and higher stacks, developed by Jacob Lurie and Bertrand Toën, extend applications to homotopical and derived moduli problems appearing in Kontsevich's deformation quantization and Toen's higher-categorical frameworks.

Technical foundations and descent theory

Foundations rely on sites, topologies, and descent formalism of Alexander Grothendieck and Jean Giraud, together with 2-categorical language by Jean Bénabou and representability and approximation theorems by Michael Artin. Key technical tools include the cotangent complex of Luc Illusie, Artin's criteria for algebraicity, and étale and fppf topologies used by Grothendieck and Deligne. Modern treatments incorporate derived algebraic geometry by Jacob Lurie, obstruction theories by Behrend and Fantechi, and stack-theoretic enhancements by Laumon and Moret-Bailly, providing a rigorous toolkit for constructing and analyzing stacks in concrete moduli problems.

Category:Algebraic geometry