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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
Abstract homotopical model for topological spaces
In category theory, a branch of mathematics, an ∞-groupoid is an abstract homotopical model for topological spaces. One model uses Kan complexes which
∞-groupoid
Algebraic structure with a binary operation
In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with
Magma_(algebra)
Constructs in non-commutative geometry
mathematics, a quantum groupoid is any of a number of notions in noncommutative geometry analogous to the notion of groupoid. In usual geometry, the
Quantum_groupoid
In algebraic topology, the fundamental groupoid is a certain topological invariant of a topological space. It can be viewed as an extension of the more
Fundamental_groupoid
In mathematics, an action groupoid or a transformation groupoid is a groupoid that expresses a group action. Namely, given a (right) group action X ×
Action_groupoid
Algebraic structure
In abstract algebra, a central groupoid is an algebraic structure defined by a binary operation ⋅ {\displaystyle \cdot } on a set of elements that satisfies
Central_groupoid
Internal groupoid in the category of smooth manifolds
In mathematics, a Lie groupoid is a groupoid where the set Ob {\displaystyle \operatorname {Ob} } of objects and the set Mor {\displaystyle \operatorname
Lie_groupoid
In category theory, a branch of mathematics, a groupoid object is both a generalization of a groupoid which is built on richer structures than sets, and
Groupoid_object
In mathematics, the concept of groupoid algebra generalizes the notion of group algebra. Given a groupoid ( G , ⋅ ) {\displaystyle (G,\cdot )} (in the
Groupoid_algebra
Set endowed with a partial binary operation
partial groupoid (also called halfgroupoid, pargoid, or partial magma) is a set endowed with a partial binary operation. A partial groupoid is a partial
Partial_groupoid
Mathematical structure in differential geometry
{\displaystyle T^{*}M} is not always integrable to a Lie groupoid. A symplectic groupoid is a Lie groupoid G ⇉ M {\displaystyle {\mathcal {G}}\rightrightarrows
Poisson_manifold
Generalized manifold
diffeomorphisms. An orbifold groupoid is given by one of the following equivalent definitions: a proper étale Lie groupoid; a proper Lie groupoid whose isotropies
Orbifold
Groupoid related to the Mathieu group M12
In mathematics, the Mathieu groupoid M13 is a groupoid acting on 13 points such that the stabilizer of each point is the Mathieu group M12. It was introduced
Mathieu_groupoid
especially in differential and algebraic geometries, an inertia stack of a groupoid X is a stack that parametrizes automorphism groups on X {\displaystyle
Inertia_stack
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
Mathematical object that generalizes the standard notions of sets and functions
this sense is called an isomorphism. A groupoid is a category in which every morphism is an isomorphism. Groupoids are generalizations of groups, group
Category_(mathematics)
Mathematical behavior near singularities
Analogous to the fundamental groupoid it is possible to get rid of the choice of a base point and to define a monodromy groupoid. Here we consider (homotopy
Monodromy
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
Concept in category theory
categories fibered in groupoids comes from groupoid objects internal to a category C {\displaystyle {\mathcal {C}}} . So given a groupoid object x ⇉ t s y
Fibred_category
Branch of mathematics
graded algebras; and constructions related to deformation quantization, groupoid C*-algebras, cyclic homology, and K-theory. A standard example is the noncommutative
Noncommutative_geometry
Hypothesis in mathematical category theory
homotopy hypothesis states, homotopy-theoretically speaking, that the ∞-groupoids are spaces. One version of the hypothesis was claimed to be proved in
Homotopy_hypothesis
Type theory in logic and mathematics
"The groupoid model refutes uniqueness of identity proofs", in which they showed that intensional type theory had a model in the category of groupoids. This
Homotopy_type_theory
Operation in group theory
{\displaystyle B(H\rtimes N)} , the (groupoid associated to) semidirect product. Another generalization is for groupoids. This occurs in topology because
Semidirect_product
Sliding puzzle with fifteen pieces and one space
transformations of the 15 puzzle form a groupoid (not a group, as not all moves can be composed); this groupoid acts on configurations. Because the combinations
15_puzzle
7-regular undirected graph with 50 nodes and 175 edges
1,0),(1,2,0),(1,3,0),(1,4,0)\}} . (Although the authors use the word "groupoid", it is in the sense of a binary function or magma, not in the category-theoretic
Hoffman–Singleton_graph
Theorem in category theory
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Lawvere's_fixed-point_theorem
Generalisation of a sheaf; a fibered category that admits effective descent
with image V. A stack is called a stack in groupoids or a (2,1)-sheaf if it is also fibered in groupoids, meaning that its fibers (the inverse images
Stack_(mathematics)
Concept in category theory
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Monoidal_functor
Mathematical concept for comparing objects
a special case of a groupoid include: Whereas the notion of "free equivalence relation" does not exist, that of a free groupoid on a directed graph does
Equivalence_relation
Generalization of category theory
studies algebraic invariants of spaces, such as the fundamental weak ∞-groupoid. In higher category theory, the concept of higher categorical structures
Higher_category_theory
Continuous function whose domain is a closed unit interval
in this category is an isomorphism, this category is a groupoid called the fundamental groupoid of X . {\displaystyle X.} Loops in this category are the
Path_(topology)
morphisms are the invertible morphisms in C. In other words, it is the largest groupoid subcategory. As a functor C ↦ core ( C ) {\displaystyle C\mapsto \operatorname
Core_of_a_category
General theory of mathematical structures
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Category_theory
Generalization of algebraic spaces or schemes
One of the motivating examples of an algebraic stack is to consider a groupoid scheme ( R , U , s , t , m ) {\displaystyle (R,U,s,t,m)} over a fixed scheme
Algebraic_stack
Infinitesimal version of Lie groupoid
of Lie groupoids that Lie algebras play in the theory of Lie groups: reducing global problems to infinitesimal ones. Indeed, any Lie groupoid gives rise
Lie_algebroid
Study of categorified structures
algebra (HDA), a double groupoid is a generalisation of a one-dimensional groupoid to two dimensions, and the latter groupoid can be considered as a special
Higher-dimensional_algebra
Generalization of a category
fundamental ∞-groupoid of X. S(X) is a quasi-category in which every morphism is invertible. The homotopy category of S(X) is the fundamental groupoid of X. More
Quasi-category
Branch of mathematics
application is also handled more simply by the use of covering morphisms of groupoids, and that technique has yielded subgroup theorems not yet proved by methods
Algebraic_topology
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
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Natural_transformation
Mapping between categories
be composed unless they share an endpoint. Thus one has the fundamental groupoid instead of the fundamental group, and this construction is functorial.
Functor
group, a groupoid has a different identity element for each object. The connection between networks and groupoid theory centers on the groupoid B G {\displaystyle
Fibration_symmetry
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
Most general completion of a commutative square given two morphisms with same domain
Brown "Topology and Groupoids" pdf available Gives an account of some categorical methods in topology, use the fundamental groupoid on a set of base points
Pushout_(category_theory)
Category in which all small limits exist
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Complete_category
English mathematician (1937–2020)
have deep connections to string theory. Conway introduced the Mathieu groupoid, an extension of the Mathieu group M12 to 13 points. As a graduate student
John_Horton_Conway
Symmetric monoidal category with a special involution
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Dagger symmetric monoidal category
Dagger_symmetric_monoidal_category
Embedding of categories into functor categories
∗ {\displaystyle *} such that every morphism is an isomorphism (i.e. a groupoid with one object). Then G = H o m C ( ∗ , ∗ ) {\displaystyle G=\mathrm {Hom}
Yoneda_lemma
Mathematical category
this gives the category of G {\displaystyle G} -sets. Similarly, for a groupoid G {\displaystyle {\mathcal {G}}} the category of presheaves on G {\displaystyle
Topos
Seminal math text
word "stacks" in the title refers to what are nowadays usually called "∞-groupoids", one possible definition of which Grothendieck sketches in his manuscript
Pursuing_Stacks
In mathematics, invertible homomorphism
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Isomorphism
Concept in differential geometry
stack over differentiable manifolds which admits an atlas, or as a Lie groupoid up to Morita equivalence. Differentiable stacks are particularly useful
Differentiable_stack
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
Algebraic topology theory
cohomology of a smooth manifold is a special example of the groupoid cohomology of a Lie groupoid. This is because given a G {\displaystyle G} -space X {\displaystyle
Equivariant_cohomology
Concept in mathematical category theory
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Symmetric_monoidal_category
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
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Lift_(mathematics)
Higher categorical generalization of a topos
colimits in X are universal, (3) coproducts in X are disjoint and (4) every groupoid object in X is effective. Mathematics portal Bousfield localization Homotopy
∞-topos
Two continuous functions can be glued together to create another continuous function
the use of piecewise functions. For example, in the book Topology and Groupoids, where the condition given for the statement below is that A ∖ B ⊆ Int
Pasting_lemma
Object in category theory
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Natural_numbers_object
Russian mathematician (born 1962)
he collaborated with Vladimir Voevodsky on ∞ {\displaystyle \infty } -groupoids, following the proposal made by Alexander Grothendieck in Esquisse d'un
Mikhail_Kapranov
Uniformity in all orientations
isotropy group is the group of isomorphisms from any object to itself in a groupoid.[dubious – discuss] An isotropy representation is a representation of an
Isotropy
Mathematics concept
normal form for the fundamental groupoid of a graph of groups. In this approach there is considerable use of free groupoids on a directed graph. Grushko's
Free_group
Algebraic structure
In abstract algebra, a medial magma or medial groupoid is a magma or groupoid (that is, a set with a binary operation) that satisfies the identity (x
Medial_magma
Five sporadic simple groups
shown that one can also extend this sequence up, obtaining the Mathieu groupoid M13 acting on 13 points. M21 is simple, but is not a sporadic simple group
Mathieu_group
Category of non-empty finite ordinals and order-preserving maps
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Simplex_category
Type of category in category theory
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Cartesian_closed_category
Generalized object in category theory
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Product_(category_theory)
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
Four-point non-Hausdorff topological space
like S1, the result follows from the groupoid Seifert-van Kampen theorem, as in the book Topology and Groupoids. More generally, McCord has shown that
Pseudocircle
Algebraic structure with a ternary operation
inverse of G. The heap of a group may be generalized again to the case of a groupoid which has two objects A and B when viewed as a category. The elements of
Heap_(mathematics)
Type of category in category theory
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Additive_category
Set with associative invertible operation
x)\simeq G} . More generally, a groupoid is any small category in which every morphism is an isomorphism. In a groupoid, the set of all morphisms in the
Group_(mathematics)
Categorical generalization of a function space in set theory
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Exponential_object
Functor type
with its unique element. A group G can be considered a category (even a groupoid) with one object which we denote by •. A functor from G to Set then corresponds
Representable_functor
Relationship between two functors abstracting many common constructions
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Adjoint_functors
Algebraic structure with an associative operation and an identity element
Required Unneeded Unneeded Small category Unneeded Required Required Unneeded Groupoid Unneeded Required Required Required Magma Required Unneeded Unneeded Unneeded
Monoid
Aspect of category theory
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Coequalizer
French mathematician (1928–2014)
Agamben and Hervé Le Tellier. Gallimard. p. 64. ISBN 978-2-07-316366-0. ∞-groupoid λ-ring AB5 category Abelian category Accessible category Algebraic geometry
Alexander_Grothendieck
Surjective homomorphism
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Epimorphism
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Localization_of_a_category
Structure in group theory (in mathematics)
but an inductive groupoid, in the sense of category theory. This close connection between inverse semigroups and inductive groupoids is embodied in the
Inverse_semigroup
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
American mathematician (born 1943)
mathematical physics, including Riemannian geometry, symplectic geometry, Lie groupoids, geometric mechanics and deformation quantization. Among his most important
Alan_Weinstein
Collection of maps which give the same result
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Commutative_diagram
Concept in mathematics
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Tensor–hom_adjunction
Bi-universal property in category theory
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Zero_morphism
Mathematical group
to its action on the Teichmüller tower of Teichmüller groupoids Tg,n, the fundamental groupoids of moduli stacks of genus g curves with n points removed
Grothendieck–Teichmüller group
Grothendieck–Teichmüller_group
the category of ∞-groupoids. Thus, the theorem can be viewed as an instance of Grothendieck's homotopy hypothesis which says ∞-groupoids are spaces (or that
Milnor's theorem on Kan complexes
Milnor's_theorem_on_Kan_complexes
Branch of mathematics
there is the notion of a fundamental groupoid (and higher variants): by definition, the fundamental groupoid of a space X is the category where the
Homotopy_theory
Overview of and topical guide to category theory
Epimorphism Monomorphism Zero morphism Normal morphism Dual (category theory) Groupoid Image (category theory) Coimage Commutative diagram Cartesian morphism
Outline_of_category_theory
Product of two categories, in category theory
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
Product_category
Graphical representation of a morphism
n-categories Bicategory (pseudofunctor) Tricategory Tetracategory Kan complex ∞-groupoid ∞-topos Strict n-categories 2-category (2-functor) 3-category Categorified
String_diagram
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
Map between simplicial sets with lifting property
{G}}} with an infinity groupoid. It is conjectured that the homotopy category of geometric realizations of infinity groupoids is equivalent to the homotopy
Kan_fibration
In particular, in higher category theory, it proves the statement "an ∞-groupoid is a Kan complex", which is a version of the homotopy hypothesis. The theorem
Joyal's extension and lifting theorems
Joyal's_extension_and_lifting_theorems
Matrix with every entry equal to one
is a matrix of ones, can be used to characterize the central groupoids. Central groupoids are algebraic structures that obey the identity ( a ⋅ b ) ⋅ (
Matrix_of_ones
Construction in algebra
HL mentioned above. For example, a finite groupoid algebra is a weak Hopf algebra. In particular, the groupoid algebra on [n] with one pair of invertible
Hopf_algebra
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
GROUPOID
GROUPOID
GROUPOID
GROUPOID
Male
French
Old French form of Latin Hubertus, HUBERT means "bright heart/mind/spirit."Â
Girl/Female
American, Australian, Christian
Combination of Rose and Anne
Boy/Male
Hindu, Indian, Traditional
Fresh Butter
Girl/Female
American, Australian, Chinese, Danish, Japanese, Latin, Swedish
The Fifth Month; May; The Youngest of Sisters; Beautiful; Plum; Enchanting; Rose; Alliance; Oath; Great One; Sprouting Life
Girl/Female
Tamil
Dawn, The earth, First light
Male
Chinese
young blue.
Male
Italian
Short form of Italian Bartolomeo, MEO means "son of Talmai."
Girl/Female
Arabic, Muslim
The Mother of Kulsum
Boy/Male
Hindu, Indian
One who is Happy in Following Dharma
Female
Greek
(ΑναστασοÏλα) Variant form of Greek Anastasios, ANASTASOULA means "resurrection."
GROUPOID
GROUPOID
GROUPOID
GROUPOID
GROUPOID