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| groupoid | |
|---|---|
| Name | groupoid |
| Type | Algebraic structure |
| Field | Mathematics |
groupoid
A groupoid is an algebraic structure generalizing groups by allowing partially defined binary operations; it appears in several areas including Algebraic Topology, Category Theory, Differential Geometry, Mathematical Physics, and Operator Algebras. Introduced in modern form by H. Brandt and systematized in later work by Charles Ehresmann, groupoids connect constructions in the theories of Lie group, fundamental group, Galois theory, Morita equivalence, and C*-algebra.
A groupoid consists of a set of arrows together with a set of objects and structure maps for source, target, multiplication (composition) defined exactly when the target of one arrow equals the source of another, identity arrows for each object, and inverses for each arrow; this axiomatization parallels definitions used in Category Theory and in the treatment of Lie groups and Étale groupoids. The algebraic data can be encoded as a small category in which every morphism is invertible, a viewpoint closely related to treatments in works by Saunders Mac Lane, Samuel Eilenberg, and Jean Bénabou.
Basic examples include the discrete groupoid on a set of objects (identity arrows only) and the pair groupoid X×X over a set X, examples often discussed alongside Évariste Galois-inspired symmetry objects and in expositions by Emmy Noether. The fundamental groupoid of a topological space generalizes the fundamental group and is central in texts by Allen Hatcher and George W. Whitehead, while action groupoids arising from a group acting on a set relate to constructions used by Sophus Lie and in the literature of Mikhail Gromov. Holonomy groupoids associated with foliations are developed in the work of Alain Connes, André Haefliger, and Philippe Tondeur.
Composition in a groupoid is associative where defined, and every arrow has a two-sided inverse, mirroring axioms familiar from Emmy Noether-inspired algebraic frameworks and from study of Arthur Cayley's permutation representations. Isotropy groups at objects are true groups and connect to stabilizer subgroups in Frobenius-style representation theory; orbit spaces and orbit equivalence reflect themes from David Hilbert's invariant theory and from modern work on Morita equivalence by Paul Baum and Alain Connes. Quotient constructions for groupoids can produce stacks and gerbes featured in expositions by Alexander Grothendieck, Jean Giraud, and Gérard Laumon.
Viewed as a small category with all morphisms invertible, groupoids play roles in categorical formulations by Saunders Mac Lane and F. William Lawvere, and relate to internal groupoids in a topos as studied by Grothendieck and André Joyal. Nerve constructions produce simplicial sets used in the homotopical treatments of Daniel Quillen and Vladimir Voevodsky, while classifying spaces for groupoids link to the work of Graeme Segal and John Milnor on classifying complexes. Adjunctions, equivalences, and 2-categorical enhancements involve researchers such as Ross Street and Max Kelly.
Adding topology yields topological groupoids and, with smooth structure, Lie groupoids central to theories of Élie Cartan and Sophus Lie. Étale groupoids are important in the study of C*-algebras by Jean Renault and in noncommutative geometry by Alain Connes; holonomy and monodromy groupoids appear in foliation theory developed by Alan Weinstein and Miguel \'Angela Crainic. Lie algebroids form the infinitesimal counterpart, studied by Kirill Mackenzie and Marius Crainic, analogous to the Lie algebra–Lie group relationship prominent in treatments by Nicolas Bourbaki.
Groupoids model symmetry beyond global group actions and appear in orbifold theory and stack theory used by W. Thurston and Edward Witten. In algebraic geometry, groupoids present algebraic stacks in the style of Deligne–Mumford and Artin; moduli problems treated by Pierre Deligne and David Mumford use groupoid presentations. In mathematical physics, groupoids underpin quantization approaches, path integral symmetries, and gauge theories discussed by Michael Atiyah, Isadore Singer, and Gerard 't Hooft; they also structure categorical quantum field theory as in work by Graeme Segal and Jacob Lurie.
Generalizations include topological groupoids, Lie groupoids, étale groupoids, measured groupoids in ergodic theory associated with Alfredo N. Koerner-style analysis, and higher groupoids central to homotopy type theory and higher category programs led by Vladimir Voevodsky and Jacob Lurie. Related constructs include groupoid C*-algebras studied by Jean Renault, stacks and gerbes in algebraic geometry by Jean Giraud and Alexander Grothendieck, and bicategories of fractions appearing in literature by Gabriel Zisman and Paul Roberts.
Category:Algebraic structures