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a double groupoid generalises the notion of groupoid and of category to a higher dimension. A double groupoid D is a higher-dimensional groupoid involving
Double_groupoid
Category where every morphism is invertible; generalization of a group
homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen
Groupoid
Study of categorified structures
higher-dimensional algebra (HDA), a double groupoid is a generalisation of a one-dimensional groupoid to two dimensions, and the latter groupoid can be considered as
Higher-dimensional_algebra
Branch of mathematics
Brown, R.; Hardie, K.; Kamps, H.; Porter, T. (2002), "The homotopy double groupoid of a Hausdorff space", Theory Appl. Categories, 10 (2): 71–93. Brown
Algebraic_topology
Describes the fundamental group in terms of a cover by two open path-connected subspaces
groupoids analogous to that of the group of integers in the theory of groups. The groupoid I {\displaystyle {\mathcal {I}}} also allows for groupoids
Seifert–Van_Kampen_theorem
properties and applications of homotopy groupoids and filtered spaces. Noncommutative double groupoids and double algebroids are only the first examples
Nonabelian_algebraic_topology
Generalized manifold
to the Morita equivalence class of the unit groupoid. If N is a compact manifold with boundary, its double M can be formed by gluing together a copy of
Orbifold
starting from groupoids. These are more abstract concepts than the Lie algebroids that play a similar role in the theory of Lie groupoids to that of Lie
R-algebroid
In mathematics, particularly category theory, a 2-group is a groupoid with a way to multiply objects and morphisms, making it resemble a group. They are
2-group
change made by ChatGPT. Brown, Ronald; Spencer, Christopher B. (1976). "Double groupoids and crossed modules" (PDF). Cahiers de Topologie et Géométrie Différentielle
Elementary theory of abstract categories
Elementary_theory_of_abstract_categories
ISSN 1432-0916. Chang, Liang (2014-04-01). "Kitaev models based on unitary quantum groupoids". Journal of Mathematical Physics. 55 (4): 041703. arXiv:1309.4181. Bibcode:2014JMP
Quantum_double_model
Algebraic topology theory
_{G}^{\bullet }(X)} which is the totalization of the de-Rham double complex of the groupoid. The terms in the Cartan complex are Ω G n ( X ) = ⨁ 2 k + i
Equivariant_cohomology
Transformations induced by a mathematical group
by the action groupoid G′ = G ⋉ X associated to the group action. The stabilizers of the action are the vertex groups of the groupoid and the orbits
Group_action
Mathematical group of the homotopy classes of loops in a topological space
Kampen's Theorem: A discussion of the fundamental groupoid of a topological space and the fundamental groupoid of a simplicial set Animations to introduce fundamental
Fundamental_group
Mathematical concept
examples of bicomplexes that come up in nature. In particular, for a Lie groupoid, there is a bicomplex associated to itpg 7-8 which can be used to construct
Double_complex
English mathematician (1935–2024)
theory Seifert–van Kampen theorem Groupoids Algebraic topology Nonabelian algebraic topology R-algebroids Double groupoids Homology (mathematics) Alexander
Ronald_Brown_(mathematician)
Generalization of category
2-category with natural transformations that map to the identity. Like Cat, groupoids (categories where morphisms are invertible) form a 2-category, where a
2-category
Type of continuous map in topology
fundamental groupoid of the orbit space X/G is isomorphic to the orbit groupoid of the fundamental groupoid of X, i.e. the quotient of that groupoid by the
Covering_space
Mathematical theorem
categories of small categories or of groupoids. Instead the notion of group object in the category of groupoids turns out to be equivalent to the notion
Eckmann–Hilton_argument
Mathematical structure in non-Riemannian differential geometry
differentiated in order to obtain a Lie bialgebroid. A Poisson groupoid is a Lie groupoid G ⇉ M {\displaystyle G\rightrightarrows M} together with a Poisson
Lie_bialgebroid
Study of abstract machines and automata
automaton groupoid. Therefore, in the most general case, categories of variable automata of any kind are categories of groupoids or groupoid categories
Automata_theory
Central object of study in category theory
V^{**}} from the vector space into its double dual. These maps are "natural" in the following sense: the double dual operation is a functor, and the maps
Natural_transformation
Topological invariant in mathematics
of a finite groupoid is the sum of 1/ |Gi |, where we picked one representative group Gi for each connected component of the groupoid. Euler calculus
Euler_characteristic
Mathematical construction used in homotopy theory
iterate the procedure, and G {\displaystyle G} is homotopy equivalent to the double loop space Ω 2 B ( B G ) {\displaystyle \Omega ^{2}B(BG)} . In case G {\displaystyle
Simplicial_set
Mathematical connection between field theory and group theory
Galois theory of Grothendieck, and some generalisations, leading to Galois groupoids.) Lang, Serge (1994). Algebraic Number Theory. Berlin, New York: Springer-Verlag
Galois_theory
Research program on the symmetries of geometry
ISBN 978-1-4020-9383-8 Jean Pradines, In Ehresmann's footsteps: from group geometries to groupoid geometries (English summary) Geometry and topology of manifolds, 87–157
Erlangen_program
category. groupoid 1. A category is called a groupoid if every morphism in it is an isomorphism. 2. An ∞-category is called an ∞-groupoid if every morphism
Glossary_of_category_theory
Symbols for constants, special functions
factorial the complete elliptic integral of the third kind the fundamental groupoid osmotic pressure π {\displaystyle \pi } represents: Archimedes' constant
Greek letters used in mathematics, science, and engineering
Greek_letters_used_in_mathematics,_science,_and_engineering
Derivative used in gauge theories
This leads to the idea of modding out the gauge group to obtain the gauge groupoid as the closest description of the gauge connection in quantum field theory
Gauge_covariant_derivative
Variant of the notion of the center of a monoid, group, or ring to a category
1090/S0894-0347-10-00669-7, MR 2669705, S2CID 2202294 Hinich, Vladimir (2007), "Drinfeld double for orbifolds", Israel mathematical conference proceedings. Quantum groups
Center_(category_theory)
*-algebra of bounded operators on a Hilbert space
Neumann algebras of a measurable equivalence relation and a measurable groupoid can be defined. These examples generalise von Neumann group algebras and
Von_Neumann_algebra
French mathematician (1869–1951)
not always possible, the set of transformations is not a group (but a groupoid in modern terminology), thus the name pseudogroup. Cartan considered only
Élie_Cartan
Doubling map on the unit interval
The fractal eigenfunctions show an explicit symmetry under the fractal groupoid of the modular group; this is developed in greater detail in the article
Dyadic_transformation
Group that is also a differentiable manifold with group operations that are smooth
also to a different generalization of Lie groups, namely Lie groupoids, which are groupoid objects in the category of smooth manifolds with a further requirement
Lie_group
theoretical physicists and mathematicians focused on quantum group and quantum groupoid applications in quantum theories; the proceedings of the meeting are published
Timeline_of_quantum_mechanics
Semigroup in abstract algebra
e = ett* for some projection e. In a *-semigroup, PI(S) is an ordered groupoid with the partial product given by s⋅t = st if s*s = tt*. In terms of examples
Semigroup_with_involution
Michigan. Retrieved May 10, 2017. Certaine, Jeremiah (1945). Lattice-ordered groupoids and some related problems. Cambridge, Mass: Harvard University. Retrieved
List of African-American mathematicians
List_of_African-American_mathematicians
Subject area in mathematics
fixed zero-object 0 {\displaystyle 0} . Note the classifying space of a groupoid B G {\displaystyle B{\mathcal {G}}} moves the homotopy groups up one degree
Algebraic_K-theory
Mathematics glossary
X , x 0 ) {\displaystyle \pi _{1}(X,x_{0})} . fundamental groupoid The fundamental groupoid of a space X is the category whose objects are the points
Glossary of algebraic topology
Glossary_of_algebraic_topology
British quantum physicist (1935–2025)
ISBN 978-3-540-70622-9. Hiley, B.J. (2010). "Process, Distinction, Groupoids and Clifford Algebras: An Alternative View of the Quantum Formalism" (PDF)
Basil_Hiley
American mathematician and novelist
2014-03-09, retrieved 2014-01-02 Schneps, Leila (2003), "Fundamental groupoids of genus zero moduli spaces and braided tensor categories", Moduli Spaces
Leila_Schneps
model featured in Giraud's theory of gerbes, which are roughly sheaves of groupoids over M. In 1994 Murray introduced bundle gerbes, which are geometric realizations
Bundle_gerbe
Generalization of associativity properties
include, for example, modules over a commutative ring, chain complexes, groupoids (or even the category of categories itself), coalgebras, etc. Given a
Operad
German mathematician
Ralph M., Khlebnikov, Sergei and Wehefritz-Kaufmann, Birgit. "Re-gauging groupoid, symmetries and degeneracies for Graph Hamiltonians and applications to
Ralph_Kaufmann
Polish mathematician
Grabowska (11 November 2015). "Graded Bundles in the Category of Lie Groupoids". Symmetry, Integrability and Geometry: Methods and Applications. 11 (11 ed
Janusz_Grabowski
DOUBLE GROUPOID
DOUBLE GROUPOID
Surname or Lastname
English
English : nickname for a sickly person, from French debile ‘frail’, ‘weak’ (from Latin debilis).Americanized spelling of German Diebel.
Surname or Lastname
English
English : variant spelling of Dowdell.Possibly an altered spelling of German Daudel, Dautel, variants of Dietz.
Girl/Female
Christian, Hindu, Indian, Kannada
Money
Surname or Lastname
English
English : from the medieval personal name Dobbe, one of several pet forms of Robert in which the initial letter was altered. Compare Hobbs.
Girl/Female
Latin
Mistress of the home.
Surname or Lastname
English
English : possibly a variant of Goble or Gobel.Perhaps an Americanized spelling of French Gobeil.
Surname or Lastname
English
English : variant of Coble.Americanized spelling of German Kobel.
Surname or Lastname
English
English : habitational name from Wombwell in South Yorkshire, named with the Old English byname Wamba meaning ‘belly’ (or this word used in a transferred topographical sense) + Old English well(a) ‘spring’, ‘stream’.
Surname or Lastname
French
French : from a reduced form of the Germanic personal name Hildo (see Hildebrand, Houde).French : habitational name from any of several places in Normandy called La Houle or Les Houles, named in Old French with the singular or plural of houle ‘cave’.English : variant of Hole.
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : nickname from Old French doubel ‘twin’ (literally ‘double’, from Late Latin duplus, classical Latin duplex, from du(o) ‘two’ + plek, a root meaning ‘fold’).
Boy/Male
Hindu, Indian
Money; Russian Currency
Surname or Lastname
English
English : from a variant of the medieval personal name Tebald, Tibalt (see Theobald).
Boy/Male
Hindu
Born during the rainy season, Money
Surname or Lastname
English
English : of uncertain origin; perhaps derived from the vocabulary word soul as a term of affection.French (Soulé) : variant of Soulier 1.George Soule (1600–80), one of the passengers on the Mayflower in 1620, was one of the founders of Duxbury, MA, where he became comparatively wealthy. He left eight children.
Male
English
Anglicized form of Irish Gaelic Dubhghall, DOYLE means "black stranger."Â
Girl/Female
Scottish
From the Gaelic 'dubhglas' meaning dark water, dark stream, or from the dark river.
Male
English
English name derived from the vocabulary word, from Latin nobilis, NOBLE means "noble."
Surname or Lastname
English (of Norman origin)
English (of Norman origin) : variant of Double.In some cases, probably an altered spelling of South German Dobel or Döbel, a topographic name for someone who lived in a gorge or deep valley, Middle High German southern dialect tobel.
Boy/Male
British, Christian, English
Dark Water; In the Seventeenth Century; Diminutive of Douglas
Surname or Lastname
English
English : variant of Dibble.Altered spelling of German Deibel or Deubel.
DOUBLE GROUPOID
DOUBLE GROUPOID
Girl/Female
Hindu, Indian
People who Give
Girl/Female
Muslim
Beautiful and tender body
Boy/Male
Tamil
Samarthi | ஸமரà¯à®¤à¯€Â
Symbol of peace
Boy/Male
Hindu
Very famous
Boy/Male
Hindu, Indian, Marathi
King
Girl/Female
Muslim
Rose garden, Inhabited town, Flourishing
Boy/Male
Tamil
Lord of all virtues, Lord Ganesh
Boy/Male
British, English
Violent
Surname or Lastname
English
English : from a metathesized diminutive of Middle English brun ‘brown’ (see Brown).
Girl/Female
Muslim
Universe
DOUBLE GROUPOID
DOUBLE GROUPOID
DOUBLE GROUPOID
DOUBLE GROUPOID
DOUBLE GROUPOID
n.
That which is doubled over or together; a doubling; a plait; a fold.
n.
The state of being double or doubled.
n.
A game between two pairs of players; as, a first prize for doubles.
v. i.
To set up a word or words a second time by mistake; to make a doublet.
adv.
Twice; doubly.
imp. & p. p.
of Double
n.
The act of one that doubles; a making double; reduplication; also, that which is doubled.
a.
To increase by adding an equal number, quantity, length, value, or the like; multiply by two; to double a sum of money; to double a number, or length.
n.
Double beer; strong beer.
adv.
In twice the quantity; to twice the degree; as, doubly wise or good; to be doubly sensible of an obligation.
v. t.
To load with a double charge, as of gunpowder.
adv.
In a double degree; doubly.
n.
One who, or that which, doubles.
a.
Double; doubled; reduplicative; repeated.
n.
Double-quick time, step, or march.
n.
Among compositors, a doublet (see Doublet, 2.); among pressmen, a sheet that is twice pulled, and blurred.
a.
To be the double of; to exceed by twofold; to contain or be worth twice as much as.
v. t.
To double the natural darkness of (a place).