Groupoid - Citizendia

This article is about groupoids in category theory. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets For the algebraic structure with a single binary operation see magma (algebra). In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure.

In mathematics, especially in category theory and homotopy theory, a groupoid is a simultaneous generalisation of a group, a setoid (a set equipped with an equivalence relation), and a $G$ -set (a set equipped with an action of a group $G$ ). Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, a setoid is a set (or type) equipped with an Equivalence relation. In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. Groupoids are often used to capture information about geometrical objects such as manifolds. Geometry ( Greek γεωμετρία; geo = earth metria = measure is a part of Mathematics concerned with questions of size shape and relative position A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be

Groupoids were first developed by Heinrich Brandt in 1926. Heinrich Brandt ( 8 November 1886, Feudingen - 9 OCtober 1954, Halle Saxony-Anhalt) was a German Mathematician Year 1926 ( MCMXXVI) was a Common year starting on Friday (link will display the full calendar of the Gregorian calendar.

1 Definitions
2 Examples
3 Relation to groups
4 Lie groupoids and Lie algebroids
5 References

Definitions

Algebraic definition

A groupoid is a set $G$ with two operations: a partially defined binary operation $\ast$ and a total (everywhere defined) function $- 1$ , which satisfy the following three conditions on elements $f$ and $g$ of $G$ :

Associativity: For all $a$ , $b$ and $c$ in $G$ , $(a \ast b) \ast c = a \ast (b \ast c)$ , if either product is defined. Domain of a partial function There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function In Mathematics, a binary operation is a calculation involving two Operands, in other words an operation whose Arity is two The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function In Mathematics, associativity is a property that a Binary operation can have
Identity: Where $f\ast g$ is defined $f\ast g\ast g^{-1} = f$ and $f^{-1}\ast f\ast g = g$ , uniquely. In Mathematics, the term identity has several different important meanings An identity is an equality that remains true regardless of the values of
Inverse: $f^{-1}\ast f$ and $f\ast f^{-1}$ are always defined.

Category theory definition

From a more abstract point of view, a groupoid is simply a small category in which every morphism is an isomorphism (that is, invertible). In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective To be explicit, a groupoid $G$ is:

a set $G 0$ of objects;
for each pair of objects $x$ and $y$ in $G 0$ , a set $G (x, y)$ of morphisms (or arrows) from $x$ to $y$ — we write $f : x \to y$ to indicate that $f$ is an element of $G (x, y)$ ;

equipped with:

an element $id x$ of $G (x, x)$ ;
for each triple of objects $x$ , $y$ , and $z$ , a binary function $comp x, y, z$ from $G (x, y)$ $\times$ $G (y, z)$ to $G (x, z)$ — we write $g f$ for $comp x, y, z (f, g)$ , where $f$ $\in$ $G (x, y)$ , $g$ $\in$ $G (y, z)$ ;
a function $inv x, y$ from $G (x, y)$ to $G (y, x)$ ;

such that:

if $f : x \to y$ , then $f id x = f$ and $id y f = f$ ;
if $f : x \to y$ , $g : y \to z$ , and $h : z \to w$ , then $(h g) f = h (g f)$ ;
if $f : x \to y$ , then $f inv(f) = id y$ and $inv(f) f = id x$ . In Mathematics, a binary function, or function of two variables, is a function which takes two inputs The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

Comparison of the definitions

The relation between these definitions is as follows: Given a groupoid in the category-theoretic sense, let $G$ be the disjoint union of all of the sets $G (x, y)$ , then $comp$ and $inv$ become partially defined operations on $G$ , and $inv$ will in fact be defined everywhere; so we define $\ast$ to be $comp$ and $- 1$ to be $inv$ . In Set theory, a disjoint union (or discriminated union) is a modified union operation which indexes the elements according to which set they originated This gives a groupoid in the algebraic definition. Explicit reference to $G 0$ (and hence to $id$ ) can be dropped.

On the other hand, given a groupoid in the algebraic sense, let $G 0$ be the set of all elements of the form $f\ast f^{-1}$ for elements $f$ of $G$ . In other words, the objects are identified with the identity morphisms, and $id x$ is just $x$ . Let $G (x, y)$ be the set of all elements $f$ such that $y f x$ is defined. Then $- 1$ and $\ast$ break up into several functions on the various $G (x, y)$ , which may be called $inv$ and $comp$ , respectively.

While we have referred to sets in the definitions above, one may instead want to use classes, in the same way as for other categories. In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously

Examples

Linear algebra

Given a field $K$ , the general linear groupoid $GL_\ast (K)$ consists of all invertible matrices with entries from $K$ , with composition given by matrix multiplication. In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Mathematics, matrix multiplication is the operation of multiplying a matrix with either a scalar or another matrix If $G = GL_\ast (K)$ , then $G 0$ contains a copy of the set of natural numbers, since there is one identity matrix of dimension $n$ for each natural number $n$ , although $G 0$ contains other matrices. In Mathematics, a natural number (also called counting number) can mean either an element of the set (the positive Integers or an In Linear algebra, the identity matrix or unit matrix of size n is the n -by- n Square matrix with ones on the Main $G (m, n)$ is empty unless $m = n$ , in which case it is the set of $n$ by $n$ invertible matrices. In Mathematics, and more specifically Set theory, the empty set is the unique set having no ( Zero) members

Topology

Start with a topological space $X$ and let $G 0$ be the set $X$ . Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. The morphisms from the point $p$ to the point $q$ are equivalence classes of continuous paths from $p$ to $q$ , with two paths being considered equivalent if they are homotopic. In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function In Mathematics, a path in a Topological space X is a continuous map f from the Unit interval I = to In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical Two such morphisms are composed by first following the first path, then the second; the homotopy equivalence guarantees that this composition is associative. In Mathematics, associativity is a property that a Binary operation can have This groupoid is called the fundamental groupoid of $X$ , denoted $\sqcap_1(X)$ . In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology.

An important extension of this idea is to consider the fundamental groupoid $\sqcap_1(X,A)$ where $A$ is (usually) a subset of $X$ . Here $A$ is considered as a set of base points chosen according to the geometry of the situation at hand.

Equivalence relation

If $X$ is a set and $\sim$ is an equivalence relation on $X$ , then we can form a groupoid representing this equivalence relation as follows: The objects are the elements of $X$ , and for any two elements $x$ and $y$ in $X$ , there is a single morphism from $x$ to $y$ if and only if $x\sim y$ . In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" ↔

Group action

If the group $G$ acts on the set $X$ , then we can form a groupoid representing this group action as follows: The objects are the elements of $X$ , and for any two elements $x$ and $y$ in $X$ , there is a morphism from $x$ to $y$ for every element $g$ of $G$ such that $g . In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. x = y$ . Composition of morphisms is given by the group operation in $G$ . Another way to describe $G$ -sets is the functor category $[Gr,Set]$ , where $Gr$ is the groupoid (category) with one element and isomorphic to the group $G$ . Indeed, every functor $F$ of this category defines a set $X = F (Gr)$ and for every $g$ in $G$ (i. e. morphism in $Gr$ ) induces a bijection $F_g:X\to X$ . The categorical structure of the functor $F$ assures us that $F$ defines a $G$ -action on the set $X$ . The (unique) representable functor $F:\mathrm{Gr}\to \mathrm{Set}$ is the Cayley Representation of $G$ . In fact, this functor is isomorphic to $Hom(Gr, - )$ and so sends $ob(Gr)$ to the set $Hom(Gr,Gr)$ which is by definition the "set" $G$ and the morphism $g$ of $Gr$ (i. e. the element $g$ of $G$ ) to the permutation $F g$ of the set $G$ . We deduce from the Yoneda Embedding that the group $G$ is isomorphic to the group $\{F_g \mid g \in G\}$ which is a subgroup of the group of permutations of $G$ .

Fifteen puzzle

The symmetries of the Fifteen puzzle form a groupoid (not a group, as not all moves can be composed). The n -puzzle is known in various versions including the 8 puzzle, the 15 puzzle, and with various names This groupoid acts on configurations. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.

Relation to groups

	Totality	Associativity	Identity	Division
Group-like structures
Group	Yes	Yes	Yes	Yes
Monoid	Yes	Yes	Yes	No
Semigroup	Yes	Yes	No	No
Loop	Yes	No	Yes	Yes
Quasigroup	Yes	No	No	Yes
Magma	Yes	No	No	No
Groupoid	No	Yes	Yes	Yes
Category	No	Yes	Yes	No

If a groupoid has only one object, then the set of its morphisms forms a group. Domain of a partial function There are two distinct meanings in current mathematical usage for the notion of the domain of a partial function In Mathematics, associativity is a property that a Binary operation can have In Mathematics, an identity element (or neutral element) is a special type of element of a set with respect to a Binary operation on that In Mathematics, especially in elementary Arithmetic, division is an arithmetic operation which is the inverse of Multiplication. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation In Mathematics, a semigroup is an Algebraic structure consisting of a nonempty set S together with an Associative Binary operation In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division In Mathematics, especially in Abstract algebra, a quasigroup is an Algebraic structure resembling a group in the sense that " division In Abstract algebra, a magma (or groupoid) is a basic kind of Algebraic structure. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element Using the algebraic definition, such a groupoid is literally just a group. Many concepts of group theory can be generalized to groupoids, with the notion of group homomorphism being replaced by that of functor. Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function In Category theory, a branch of Mathematics, a functor is a special type of mapping between categories

If $x$ is an object of the groupoid $G$ , then the set of all morphisms from $x$ to $x$ forms a group $G (x)$ . If there is a morphism $f$ from $x$ to $y$ , then the groups $G (x)$ and $G (y)$ are isomorphic, with an isomorphism given by mapping $g$ to $f g f - 1$ . In Abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in

Every connected groupoid (that is, one in which any two objects are connected by at least one morphism) is isomorphic to a groupoid of the following form: Pick a group $G$ and a set (or class) $X$ . In Category theory, a branch of Mathematics, a connected category is a category in which for every two objects X and Y there is a In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously Let the objects of the groupoid be the elements of $X$ . For elements $x$ and $y$ of $X$ , let the set of morphisms from $x$ to $y$ be $G$ . Composition of morphisms is the group operation of $G$ . If the groupoid is not connected, then it is isomorphic to a disjoint union of groupoids of the above type (possibly with different groups $G$ per connected component). In Set theory, a disjoint union (or discriminated union) is a modified union operation which indexes the elements according to which set they originated Thus, any groupoid may be given (up to isomorphism) by a set of ordered pairs $(X, G)$ . In Mathematics, the phrase " up to xxxx" indicates that members of an Equivalence class are to be regarded as a single entity for some purpose In Mathematics, an ordered pair is a collection of two distinguishable objects one of which is identified as the first coordinate (or the first entry

Note that the isomorphism described above is not unique, and there is no natural choice. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal Choosing such an isomorphism for a connected groupoid essentially amounts to picking one object $x 0$ , a group isomorphism $h$ from $G (x 0)$ to $G$ , and for each $x$ other than $x 0$ a morphism in $G$ from $x 0$ to $x$ .

In category-theoretic terms, each connected component of a groupoid is equivalent (but not isomorphic) to a groupoid with a single object, that is, a single group. In Category theory, an abstract branch of Mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are In Category theory, two categories C and D are isomorphic if there exist Functors F: C &rarr D and G Thus any groupoid is equivalent to a multiset of unrelated groups. In Mathematics, a multiset (or bag) is a generalization of a set. In other words, for equivalence instead of isomorphism, you don’t have to specify the sets $X$ , only the groups $G$ .

Consider the examples in the previous section. The general linear groupoid is both equivalent and isomorphic to the disjoint union of the various general linear groups $G L n (F)$ . In Mathematics, the general linear group of degree n is the set of n × n invertible matrices, together with the operation On the other hand, the fundamental groupoid of $X$ is equivalent to the collection of the fundamental groups of each path-connected component of $X$ , but for an isomorphism you must also specify the set of points in each component. In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. In Topology and related branches of Mathematics, a connected space is a Topological space which cannot be represented as the disjoint union of The set $X$ with the equivalence relation $\sim$ is equivalent (as a groupoid) to one copy of the trivial group for each equivalence class, but for an isomorphism you must also specify what each equivalence class is. In Mathematics, a trivial group is a group consisting of a single element In Mathematics, given a set X and an Equivalence relation ~ on X, the equivalence class of an element a in X Finally, the set $X$ equipped with an action of the group $G$ is equivalent (as a groupoid) to one copy of $G$ for each orbit of the action, but for an isomorphism you must also specify what set each orbit is. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups.

The collapse of a groupoid into a mere collection of groups loses some information, even from a category-theoretic point of view, because it’s not natural. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal Thus when groupoids arise in terms of other structures, as in the above examples, it can be helpful to maintain the full groupoid. If you don’t, then you must choose a way to view each $G (x)$ in terms of a single group, and this can be rather arbitrary. In our example from topology, you would have to make a coherent choice of paths (or equivalence classes of paths) from each point $p$ to each point $q$ in the same path-connected component.

As a more illuminating example, the classification of groupoids with one endomorphism does not reduce to purely group theoretic considerations. This is analogous to the fact that the classification of vector spaces with one endomorphism is non trivial.

Morphisms of groupoids come in more kinds than those of groups: we have, for example, fibrations, covering morphisms, universal morphisms, quotient morphisms. Thus a subgroup $H$ of a group $G$ yields an action of $G$ on the set of cosets of $H$ in $G$ and hence a covering morphism $p$ from say $K$ to $G$ where $K$ is a groupoid with vertex groups isomorphic to $H$ . In this way, presentations of the group $G$ can be lifted to presentations of the groupoid $K$ , and this is a useful way of obtaining information on presentations of the subgroup $H$ . For further information, see the books by Higgins and by Brown listed below.

Another useful fact is that the category of groupoids, unlike that of groups, is cartesian closed. In Category theory, a category is cartesian closed if roughly speaking any Morphism defined on a product of two objects can be naturally identified with a morphism

Lie groupoids and Lie algebroids

When studying geometrical objects, the arising groupoids often carry some differentiable structure, turning them into Lie groupoids. In Mathematics, an n -dimensional differential structure (or differentiable structure on a set M makes it into an n -dimensional Differential In Mathematics, a Lie groupoid is a Groupoid where the set Ob of objects and the set Mor of morphisms are both manifolds the source and These can be studied in terms of Lie algebroids, in analogy to the relation between Lie groups and Lie algebras. In Mathematics, Lie algebroids serve the same role in the theory of Lie groupoids that Lie algebras serve in the theory of Lie groups: reducing global problems In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie

References

Ronald Brown, From groups to groupoids: a brief survey, Bull. LMS, 19 (1987) 113-134, gives some of the history of groupoids, namely the origins in work of Brandt on quadratic forms, and an indication of later work up to 1987, with 160 references. These have been updated slightly in the downloadable version, available as [1]

Alan Weinstein, Groupoids: unifying internal and external symmetry, available as Groupoids.ps or weinstein.pdf
Part VI of Geometric Models for Noncommutative Algebras, by A. Cannas da Silva and A. Weinstein PDF file.
Higher dimensional group theory is a web article with lots of references explaining how the groupoid concept has to led to notions of higher dimensional groupoids, not available in group theory, with applications in homotopy theory and in group cohomology.

General theory of Lie groupoids and Lie algebroids, K.C.H. Mackenzie, CUP, 2005

Topology and groupoids, Ronald Brown, Booksurge 2006 revised and extended edition of a book previously published in 1968 and 1988. e-version available.

Categories and groupoids, P.J. Higgins, downloadable reprint of van Nostrand Notes in Mathematics, 1971, which deal with applications of groupoids in group theory and topology.

Galois theories, F. Borceux, G. Janelidze, CUP, 2001 shows how generalisations of Galois theory lead to Galois groupoids.

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